Dual-primal Finite Element Tearing Research Articles

Efficient 3D Frequency Semi-Airborne Electromagnetic Modeling Based on Domain Decomposition

Landslides are common geological hazards that often result in significant casualties and economic losses. Due to their occurrence in complex terrain areas, conventional geophysical techniques face challenges in early detection and warning of landslides. Semi-airborne electromagnetic (SAEM) technology, utilizing unmanned aerial platforms for rapid unmanned remote sensing, can overcome the influence of complex terrain and serve as an effective approach for landslide detection and monitoring. In response to the low computational efficiency of conventional semi-airborne EM 3D forward modeling, this study proposes an efficient forward modeling method. To handle arbitrarily complex topography and geologic structures, the unstructured tetrahedron mesh is adopted to discretize the earth. Based on the vector finite element formula, the Dual–Primal Finite Element Tearing and Interconnecting (FETI-DP) method is further applied to enhance modeling efficiency. It obtains a reduced order subsystem and avoids directly solving huge overall linear equations by converting the entirety problem into the interface problem. We check our algorithm by comparing it with 1D semi-analytical solutions and the conventional finite element method. The numerical experiments reveal that the FETI-DP method utilities less memory and exhibits higher computation efficiency than the conventional methods. Additionally, we compare the electromagnetic responses with the topography model and flat earth model. The comparison results indicate that the effect of topography cannot be ignored. Further, we analyze the characteristic of electromagnetic responses when the thickness of the aquifer changes in a landslide area. We demonstrate the effectiveness of the 3D SAEM method for landslide detection and monitoring.

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.

Globalization of Nonlinear FETI-DP Domain Decomposition Methods Using an SQP Approach

The globalization of Nonlinear FETI-DP (Dual Primal Finite Element Tearing and Interconnecting) methods is considered using a Sequential Quadratic Programming (SQP) approach. Nonlinear FETI-DP methods are parallel iterative solution methods for nonlinear finite element problems, based on divide and conquer, using Lagrange multipliers. In these methods, nonlinear elimination is an important ingredient to increase the convergence radius of Newton’s method. We prove standard globalization results for SQP-based globalization of Nonlinear FETI-DP, first for the case that the elimination set is empty. We then show how to combine nonlinear elimination and SQP-based globalization. The globalization preserves the block structure of the FETI-DP operator, which is the basis of the computational parallelism.Supporting numerical experiments using homogenous and heterogeneous model problems from nonlinear structural mechanics are provided. In the numerical experiments, we consider four standard choices of different elimination sets and different problem setups including stiff or almost incompressible inclusions in every subdomain. The numerical experiments illustrate that a good elimination set is important. However, the use of the SQP-based globalization approach presented here can improve the convergence of Nonlinear FETI-DP methods further, especially, if combined with a good choice of the elimination set.

Parallel Newton--Krylov BDDC and FETI-DP Deluxe Solvers for Implicit Time discretizations of the Cardiac Bidomain Equations

Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A finite element discretization in space and a segregated implicit discretization in time, based on decoupling the PDEs from the ODEs, yields at each time step the solution of a non-linear algebraic system. The Jacobian linear system at each Newton iteration is solved by a Krylov method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the recently introduced {\em deluxe} scaling of the dual variables. A polylogarithmic convergence rate bound is proven for the resulting parallel Bidomain solvers. Extensive numerical experiments on linux clusters up to two thousands processors confirm the theoretical estimates, showing that the proposed parallel solvers are scalable and quasi-optimal.

An Unconditionally Stable and Energy-Preserving Domain-Decomposition Method for Transient Modeling of Large-Scale Electromagnetic Problems

The purpose of this contribution is to introduce a dual-primal finite-element tearing and interconnecting (DP-FETI) method for the time-domain simulation of large electromagnetic problems. The most distinctive feature of the proposed method is that the continuity of the first-time derivative of the electric field intensity is enforced across the interfaces, unlike the existing formulations in electromagnetics. A theoretical proof is provided to demonstrate that this transmission condition guarantees unconditional stability and energy preservation of the method. Since the condition number of the global interface problem is greatly reduced compared to the standard finite-element time-domain (FETD) method, an iterative solver converges in far fewer iterations. This significantly reduces the simulation time, particularly when implemented on parallel computers. It is numerically shown that the method is scalable with respect to the number of subdomains and the time-step size. The numerical results for simulation of a cavity and bandgap devices are presented to demonstrate capability, accuracy, and efficiency of the proposed time-domain DP-FETI (TD-DP-FETI).

BDDC and FETI-DP algorithms with a change of basis formulation on adaptive primal constraints

BDDC (Balancing Domain Decomposition by Constraints) and FETI-DP (Dual-Primal Finite Element Tearing and Interconnecting) algorithms with adaptively enriched primal constraints are considered. The coarse component of the two algorithms is built on the set of primal unknowns consisting of those at subdomain vertices and those from the adaptive primal constraints after a change of basis. For the FETI-DP algorithm, a more general form of a preconditioner is proposed to extend the algorithm to the set of primal unknowns including those from the adaptive primal constraints. In addition, it can be shown that the two algorithms share the same spectra except those equal to one or zero when the same set of adaptive primal constraints are employed. Numerical results are included for both two and three dimensional model problems.

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.

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.

Adaptive Coarse Spaces for FETI-DP in Three Dimensions

An adaptive coarse space approach including a condition number bound for dual primal finite element tearing and interconnecting (FETI-DP) methods applied to three dimensional problems with coefficient jumps inside subdomains and across subdomain boundaries is presented. The approach is based on a known adaptive coarse space approach enriched by a small number of additional local edge eigenvalue problems. These edge eigenvalue problems serve to make the method robust and permit a condition number bound which depends only on the tolerance of the local eigenvalue problems and some properties of the domain decomposition. The introduction of the edge eigenvalue problems thus turns a well-known condition number indicator for FETI-DP and balancing domain decomposition by constraints (BDDC) methods into a condition number estimate. Numerical results are presented for linear elasticity and heterogeneous materials supporting our theoretical findings. The problems considered include those with random coefficients an...

Efficient combination of a 3D Quasi-Newton inversion algorithm and a vector dual-primal finite element tearing and interconnecting method

A Quasi-Newton method for reconstructing the constitutive parameters of three-dimensional (3D) penetrable scatterers from scattered field measurements is presented. This method is adapted for handling large-scale electromagnetic problems while keeping the memory requirement and the time flexibility as low as possible. The forward scattering problem is solved by applying the finite-element tearing and interconnecting full-dual-primal (FETI-FDP2) method which shares the same spirit as the domain decomposition methods for finite element methods. The idea is to split the computational domain into smaller non-overlapping sub-domains in order to simultaneously solve local sub-problems. Various strategies are proposed in order to efficiently couple the inversion algorithm with the FETI-FDP2 method: a separation into permanent and non-permanent subdomains is performed, iterative solvers are favorized for resolving the interface problem and a marching-on-in-anything initial guess selection further accelerates the process. The computational burden is also reduced by applying the adjoint state vector methodology. Finally, the inversion algorithm is confronted to measurements extracted from the 3D Fresnel database.

3-D Electromagnetic Scattering Computation in Free-Space With the FETI-FDP2 Method

The electromagnetic dual-primal finite element tearing and interconnecting (FETI-DPEM) method is a nonoverlapping domain decomposition method developed for the finite element analysis of large-scale electromagnetic problems, where the corner edges are globally numbered. This paper presents an extension of the FETI-DPEM2 method, named FETI-full dual primal (FETI-FDP2), where more flexible Robin-type boundary conditions are imposed, on the inner interfaces between subdomains as well as on the corner edges, leading to a new interface problem. Its capacities are tested in the framework of a three-dimensional (3-D) free-space scattering problem, with a scattered field formulation and a computational domain truncated by perfectly mathed layers (PML). First, we compare its accuracy with respect to other FETI-DPEM2 methods and to a complete resolution of the FEM problem, thanks to a direct sparse solver. We show that the convergence of iterative solvers is affected by the presence of the PML and can be accelerated by means of a more accurate approximation, between adjacent subdomains, of the Dirichlet-to-Neumann (DtN) operator. The effectiveness of the iterative solvers are also considered for different test cases. The advantages of the proposed FETI-FDP2 method combined with the associated DtN approximation is numerically demonstrated, regardless the chosen working frequency or the iterative solvers.

Fast analysis of three‐dimensional electromagnetic problems using dual‐primal finite‐element tearing and interconnecting method combined with ℋ‐matrix technique

As a powerful variant of the dual-primal finite-element tearing and interconnecting (FETI-DP) method in electromagnetics (EM), the FETI-DPEM method shows excellent convergence properties and numerical scalability. A key factor affecting the performance of the FETI-DPEM is the direct solution of sub-domain FE systems. An efficient direct solver based on hierarchical (ℋ-) matrix technique is proposed to solve the sub-domain FE systems. ℋ-matrix formatted lower-upper (LU) decomposition with subsequent forward and backward substitution (FBS) can significantly reduce the computational costs of the direct solution with sufficient accuracy. The nested dissection technique is employed to further accelerate the ℋ-LU decomposition and the ℋ-FBS. Numerical examples demonstrate that the proposed method is very robust for the three-dimensional EM simulation.

A 3D finite element analysis of large‐scale nonlinear dynamic electromagnetic problems by harmonic balancing and domain decomposition

AbstractThe dual–primal finite element tearing and interconnecting (FETI‐DP) method is applied together with the harmonic balance method and the fixed‐point (FP) method to improve the efficiency of three‐dimensional finite element analysis of large‐scale nonlinear dynamic electromagnetic problems. The FETI‐DP method decomposes the original problem into smaller subdomains problems. Combined with parallel computing techniques, the total computation time can be reduced significantly. To account for nonlinear B‐H characteristics, the FP method is applied together with the polarization formulation. Because the FP method assumes a constant reluctivity, it decouples the systems of different harmonics. The FETI‐DP method can then be applied to speed up the simulation of each harmonic in each FP iteration. Two benchmark problems and a three‐phase inductor are simulated by the proposed method to validate the formulation and demonstrate its performance. Copyright © 2015 John Wiley & Sons, Ltd.

A FETI-DP Type Domain Decomposition Algorithm for Three-Dimensional Incompressible Stokes Equations

The FETI-DP (dual-primal finite element tearing and interconnecting) algorithms, proposed by the authors in [SIAM J. Numer. Anal., 51 (2013), pp. 1235--1253] and [Internat. J. Numer. Methods Engrg., 94 (2013), pp. 128--149] for solving incompressible Stokes equations, are extended to three-dimensional problems. A new analysis of the condition number bound for using the Dirichlet preconditioner is given. The algorithm and analysis are valid for mixed finite elements with both continuous and discontinuous pressures. An advantage of this new analysis is that the numerous coarse level velocity components, required in the previous analysis to enforce the divergence-free subdomain boundary velocity conditions, are no longer needed. This greatly reduces the size of the coarse level problem in the algorithm, especially for three-dimensional problems. The coarse level velocity space can be chosen as simple as those coarse spaces for solving scalar elliptic problems corresponding to each velocity component. Both the Dirichlet and lumped preconditioners are analyzed using the same framework in this new analysis. Their condition number bounds are proved to be independent of the number of subdomains for fixed subdomain problem size. Numerical experiments in both two and three dimensions, using mixed finite elements with both continuous and discontinuous pressures, demonstrate the convergence rate of the algorithms.

FETI-DP Methods with an Adaptive Coarse Space

A coarse space is constructed for the dual-primal finite element tearing and interconnecting (FETI-DP) domain decomposition method applied to highly heterogeneous problems by solving local generalized eigenvalue problems. For certain problems with highly varying coefficients, e.g., from multiscale simulations, the coefficient jump will appear in the condition number bound even if standard techniques such as scaling and the weighting of constraints are used. The FETI-DP theory is revisited and two central estimates are identified where the dependency on the coefficient contrast can enter the condition number bound. The first is a Poincaré inequality and the second an extension theorem. These estimates are replaced by local eigenvalue problems. Enriching the FETI-DP coarse space by a few numerically computed eigenvectors yields independence of the contrast of the coefficients even in challenging situations.

A preconditioned dual–primal finite element tearing and interconnecting method for solving three-dimensional time-harmonic Maxwell's equations

A new preconditioned dual–primal nonoverlapping domain decomposition method is proposed for the finite element solution of three-dimensional large-scale electromagnetic problems. With the aid of two Lagrange multipliers, the new method converts the original volumetric problem to a surface problem by using a higher-order transmission condition at the subdomain interfaces to significantly improve the convergence of the iterative solution of the global interface equation. Similar to the previous version, a global coarse problem related to the degrees of freedom at the subdomain corner edges is formulated to propagate the residual error to the whole computational domain at each iteration, which further increases the rate of convergence. In addition, a fully algebraic preconditioner based on matrix splitting is constructed to make the proposed domain decomposition method even more robust and scalable. Perfectly matched layers (PMLs) are considered for the boundary truncation when solving open-region problems. The influence of the PML truncation on the convergence performance is investigated by examining the convergence of the transmission condition for an interface inside the PML. Numerical examples including wave propagation and antenna radiation problems truncated with PMLs are presented to demonstrate the validity and the capability of this method.

A Finite-Element-Based Domain Decomposition Method for Efficient Simulation of Nonlinear Electromechanical Problems

The dual-primal finite-element tearing and interconnecting (FETI-DP) method is combined with the Newton-Raphson method to expand the capability and improve the efficiency of 3-D finite-element analysis (FEA) of nonlinear electromechanical problems. Despite its modeling capability and high degree of accuracy, FEA has high computational complexity, especially for nonlinear analysis. The FETI-DP method is a robust domain decomposition method, which has been enhanced and applied to solve electromechanical problems involving linear materials. In this paper, the FETI-DP method is extended with the Newton-Raphson method to address problems involving nonlinearity and saturation. Using parallel computing techniques, the total computation time is reduced significantly. Linear and nonlinear regions are separated using the FETI-DP method. This further improves simulation efficiency and flexibility. Cubic splines and relaxation techniques are adopted to ensure stable and fast convergence of the Newton-Raphson method. The performance of the proposed method is compared with infolytica's MagNet, a commercial 3-D FEA solver.

A Hybrid Conformal/Nonconformal Domain Decomposition Method for Multi-Region Electromagnetic Modeling

This paper presents a unified framework for solving large-scale multi-region electromagnetic problems using a hybrid conformal/nonconformal domain decomposition method. In this method, an original electrically large problem is first divided into several regions based on its geometry. Each region is then meshed and further decomposed into smaller subdomains. As such, the decomposed problem is mesh-conformal and geometry-conformal within each region but could be mesh-nonconformal and geometry-nonconformal between two neighboring regions. To solve such a problem, a hybrid method is proposed that employs the finite-element tearing and interconnecting (FETI) method to deal with mesh-nonconformal and/or geometry-nonconformal interfaces, where a second-order transmission condition and a crosspoint correction technique are applied to improve the iterative convergence of the interface system and ensure a correct interconnection across subdomain interfaces. For mesh-conformal and geometry-conformal interfaces inside each region, the hybrid method employs the dual-primal finite element tearing and interconnecting (FETI-DP) method to construct an effective coarse grid correction for the interface problem. A unified global system of equations is finally formulated for the interface unknowns from both nonconformal and conformal interfaces. The proposed method is validated with simulation of wave propagation and radiation by broadband antennas and phased-array antennas covered with a radome and a near-field focal lens.

Fast and accurate finite element analysis of large-scale three-dimensional photonic devices with a robust domain decomposition method

A fast and accurate full-wave technique based on the dual-primal finite element tearing and interconnecting method and the second-order transmission condition is presented for large-scale three-dimensional photonic device simulations. The technique decomposes a general three-dimensional electromagnetic problem into smaller subdomain problems so that parallel computing can be performed on distributed-memory computer clusters to reduce the simulation time significantly. With the electric fields computed everywhere, photonic device parameters such as transmission and reflection coefficients are extracted. Several photonic devices, with simulation volumes up to 1.9×10(4) (λ/n(avg))3 and modeled with over one hundred million unknowns, are simulated to demonstrate the application, efficiency, and capability of this technique. The simulations show good agreement with experimental results and in a special case with a simplified two-dimensional simulation.

Nonlinear FETI-DP and BDDC Methods

New nonlinear FETI-DP (dual-primal finite element tearing and interconnecting) and BDDC (balancing domain decomposition by constraints) domain decomposition methods are introduced. In all these methods, in each iteration, local nonlinear problems are solved on the subdomains. The new approaches can significantly reduce communication and show a significantly improved performance, especially for problems with localized nonlinearities, compared to a standard Newton--Krylov--FETI-DP or BDDC approach. Moreover, the coarse space of the nonlinear FETI-DP methods can be used to accelerate the Newton convergence. It is also found that the new nonlinear FETI-DP and nonlinear BDDC methods are not as closely related as in the linear context. Numerical results for the p-Laplace operator are presented.