Finite Element Tearing And Interconnecting Methods Research Articles

Improvement of DDM Preprocessing for the Simulation of the GRAVES Radar Densified Sparse Array for Space Surveillance and Tracking

In a recent work (A. Jouade and A. Barka, “Massively parallel implementation of FETI-2LM methods for the simulation of the sparse receiving array evolution of the GRAVES radar system for space surveillance and tracking, IEEE Access , vol. 7, pp. 128968–128979, 2019), we have discussed our implementation of the finite-element tearing and interconnecting (FETI) method and the domain decomposition method (DDM) for the full-wave simulation of the large GRAVES sparse array (60 m diameter disk). Owing to the nonregular distribution of the antennas of the sparse array, such simulations in reasonable times are not accessible with existing FETI methods optimized for repetitive geometries, which do not benefit from an effective parallelization technique. A particularity of the preprocessing step of our DDM is that the complete sparse array mesh is not built upstream of the resolution of the electromagnetic problem. Each subdomain in our decomposition is equipped with its own local mesh, belonging to a limited set of unit-cell meshes, generated separately. Initially, a total of 13 692 subdomains were chosen among three types of subdomains, considering those corresponding to antennas (200), air subdomains (800), and ground plane subdomains (12 592). This simulation has required 13 692 Intel Xeon Broadwell E5-2680v4 processing cores equipped with 4 GB of memory, each handling one subdomain. A new strategy is shown, which requires only 2408 cores for the same simulation, leading to an 82.41% reduction in the required number of cores, as well as a 30% reduction in the simulation time.

Classical and all-floating FETI methods for the simulation of arterial tissues.

High-resolution and anatomically realistic computer models of biological soft tissues play a significant role in the understanding of the function of cardiovascular components in health and disease. However, the computational effort to handle fine grids to resolve the geometries as well as sophisticated tissue models is very challenging. One possibility to derive a strongly scalable parallel solution algorithm is to consider finite element tearing and interconnecting (FETI) methods. In this study we propose and investigate the application of FETI methods to simulate the elastic behavior of biological soft tissues. As one particular example we choose the artery which is - as most other biological tissues - characterized by anisotropic and nonlinear material properties. We compare two specific approaches of FETI methods, classical and all-floating, and investigate the numerical behavior of different preconditioning techniques. In comparison to classical FETI, the all-floating approach has not only advantages concerning the implementation but in many cases also concerning the convergence of the global iterative solution method. This behavior is illustrated with numerical examples. We present results of linear elastic simulations to show convergence rates, as expected from the theory, and results from the more sophisticated nonlinear case where we apply a well-known anisotropic model to the realistic geometry of an artery. Although the FETI methods have a great applicability on artery simulations we will also discuss some limitations concerning the dependence on material parameters.

Highly scalable parallel domain decomposition methods with an application to biomechanics

AbstractHighly scalable parallel domain decomposition methods for elliptic partial differential equations are considered with a special emphasis on problems arising in elasticity. The focus of this survey article is on Finite Element Tearing and Interconnecting (FETI) methods, a family of nonoverlapping domain decomposition methods where the continuity between the subdomains, in principle, is enforced by the use of Lagrange multipliers. Exact onelevel and dual‐primal FETI methods as well as related inexact dual‐primal variants are described and theoretical convergence estimates are presented together with numerical results confirming the parallel scalability properties of these methods. New aspects such as a hybrid onelevel FETI/FETI‐DP approach and the behavior of FETI‐DP for anisotropic elasticity problems are presented. Parallel and numerical scalability of the methods for more than 65 000 processor cores of the JUGENE supercomputer is shown. An application of a dual‐primal FETI method to a nontrivial biomechanical problem from nonlinear elasticity, modeling arterial wall stress, is given, showing the robustness of our domain decomposition methods for such problems.

Parallel Iterative Substructuring in Structural Mechanics

Finite Element Tearing and Interconnecting (FETI) methods are a family of nonoverlapping domain decomposition methods which have been proven to be robust and parallel scalable for a variety of elliptic partial differential equations. Here, an introduction to the classical onelevel FETI methods is given, as well as to the more recent dual-primal FETI methods and some of their variants. With the advent of modern parallel computers with thousands of processors, certain inexact components are needed in these methods to maintain scalability. An introduction to a recent class of inexact dual-primal FETI methods is presented. Scalability results for an elasticity problem using 65 536 processor cores of the JUGENE supercomputer at Forschungszentrum Julich show the potential of these methods. A hyperelastic problem from biomechanics is presented as an application of the methods to nonlinear finite element analysis.

Total FETI for contact problems with additional nonlinearities

AbstractThe paper is concerned with application of a new variant of the Finite Element Tearing and Interconnecting (FETI) method, referred to as the Total FETI (TFETI), to the solution to contact problems with additional nonlinearities. While the standard FETI methods assume that the prescribed Dirichlet conditions are inherited by subdomains, TFETI enforces both the compatibility between subdomains and the prescribed displacements by the Lagrange multipliers. If applied to the contact problems, this approach not only transforms the general nonpenetration constraints to the bound constraints, but it also generates an enriched natural coarse grid defined by the a priori known kernels of the stiffness matrices of the subdomains exhibiting rigid body modes. We combine our in a sense optimal algorithms for the solution to bound and equality constrained problems with geometric and material nonlinearities. The section on numerical experiments presents results of solution to bolt and nut contact problem with additional geometric and material nonlinear effects. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

FETI Domain Decomposition Methods for Second Order Elliptic Partial Differential Equations

AbstractA survey on certain Finite Element Tearing and Interconnecting (FETI) methods for second order elliptic partial differential equations is given. FETI methods are nonoverlapping domain decomposition algorithms where continuity across subdomain boundaries is treated, sometimes only in parts, by Lagrange multipliers. The family of FETI algorithms is among the best known and most severely tested domain decomposition methods for elliptic partial differential equations. In this article, an overview is given of the classical one‐level FETI method and the more recent dual‐primal FETI methods; both type of algorithms are considered for scalar, second order, elliptic equations and the system of linear elasticity, and are illustrated by some computational results. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Boundary Element Tearing and Interconnecting Methods

In this paper we introduce the Boundary Element Tearing and Interconnecting (BETI) methods as boundary element counterparts of the well-established Finite Element Tearing and Interconnecting (FETI) methods. In some practical important applications such as far field computations, handling of singularities and moving parts etc., BETI methods have certainly some advantages over their finite element counterparts. This claim is especially true for the sparse versions of the BETI preconditioners resp. methods. Moreover, there is an unified framework for coupling, handling, and analyzing both methods. In particular, the FETI methods can benefit from preconditioning components constructed by boundary element techniques. The first numerical results confirm the efficiency and the robustness predicted by our analysis.

On the initial estimate of interface forces in FETI methods

The balanced domain decomposition (BDD) method and the finite element tearing and interconnecting (FETI) method are two commonly used non-overlapping domain decomposition methods. Due to strong theoretical and numerical similarities, these two methods are generally considered as being equivalently efficient. However, for some particular cases, such as for structures with strong heterogeneities, FETI requires a large number of iterations to compute the solution compared to BDD. In this paper, the origin of the poor efficiency of FETI in these particular cases is traced back to bad initial estimates of the interface stresses. To improve the estimation of interface forces, a novel strategy for splitting interface forces between neighboring substructures is proposed. The additional computational cost incurred is not significant. This yields a new initialization for the FETI method and restores numerical efficiency which makes FETI comparable to BDD even for problems where FETI was performing poorly. Various simple test problems are presented to discuss the efficiency of the proposed strategy and to illustrate the so-obtained numerical equivalence between the BDD and FETI solvers.

Mechanism free domain decomposition

The simulation of three-dimensional (3D) structural dynamics on massively parallel platforms places stringent requirements on the existing software infrastructure. A constrained and nonlinear graph partitioning problem that arises in scalable iterative substructuring methods, such as finite element tearing and interconnecting (FETI) methods, is identified. New sufficient criteria on a partition are presented that ensure the applicability of FETI methods, and improve the associated preconditioner. One-dimensional finite elements in 3D structures are treated by an encapsulation method. The techniques are demonstrated on complex finite element model problems.

Dual-Primal FETI Methods for Three-Dimensional Elliptic Problems with Heterogeneous Coefficients

In this paper, certain iterative substructuring methods with Lagrange multipliers are considered for elliptic problems in three dimensions. The algorithms belong to the family of dual-primal finite element tearing and interconnecting (FETI) methods which recently have been introduced and analyzed successfully for elliptic problems in the plane. The family of algorithms for three dimensions is extended and a full analysis is provided for the new algorithms. Particular attention is paid to finding algorithms with a small primal subspace since that subspace represents the only global part of the dual-primal preconditioner. It is shown that the condition numbers of several of the dual-primal FETI methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains, the mesh size, and jumps in the coefficients. These results closely parallel those of other successful iterative substructuring methods of primal as well as dual type.

A FETI Domain Decomposition Method for Edge Element Approximations in Two Dimensions with Discontinuous Coefficients

A class of finite element tearing and interconnecting (FETI) methods for the edge element approximation of vector field problems in two dimensions is introduced and analyzed. First, an abstract framework is presented for the analysis of a class of FETI methods where a natural coarse problem, associated with the substructures, is lacking. Then, a family of FETI methods for edge element approximations is proposed. It is shown that the condition number of the corresponding method is independent of the number of substructures and grows only polylogarithmically with the number of unknowns associated with individual substructures. The estimate is also independent of the jumps of both of the coefficients of the original problem. Numerical results validating our theoretical bounds are given. The method and its analysis can be easily generalized to Raviart--Thomas element approximations in two and three dimensions.

Two-level domain decomposition methods with Lagrange multipliers for the fast iterative solution of acoustic scattering problems

We present two different but related Lagrange multiplier based domain decomposition (DD) methods for solving iteratively large-scale systems of equations arising from the finite element discretization of high-frequency exterior Helmholtz problems. The proposed methods are essentially two distinct extensions of the regularized finite element tearing and interconnecting (FETI) method to indefinite or complex problems. The first method employs a single Lagrange multiplier field to glue the local solutions at the subdomain interface boundaries. The second method employs two Lagrange multiplier fields for that purpose. The key ingredients of both of these FETI methods are the regularization of each subdomain matrix by a complex lumped mass matrix defined on the subdomain interface boundary, and the preconditioning of the global interface problem by a coarse second-level problem constructed with planar waves. We show numerically that both methods are scalable with respect to the mesh size, the subdomain size, and the wavenumber, but that the FETI method with a single Lagrange multiplier field – labeled FETI-H (H for Helmholtz) in this paper – delivers superior computational performances. We apply the FETI-H method to the parallel solution on a 24-processor Origin 2000 of an acoustic scattering problem with a submarine shaped obstacle, and report performance results that highlight the unique efficiency of this DD method for the solution of high frequency acoustic scattering problems.