Dual-primal Finite Element Tearing And Interconnecting Research Articles

Adaptive and Frugal FETI-DP for Virtual Elements

The FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) method has recently successfully been applied to virtual element discretizations, adding more flexibility to the resolution of possibly complicated underlying domain geometries. However, for second-order partial differential equations with large discontinuities in the coefficient functions, in general, the convergence rate of domain decomposition methods is known to deteriorate if the coarse space is not properly adjusted. For finite element discretizations, this problem can be solved by using adaptive coarse spaces, which guarantee a robust method for arbitrary coefficient distributions, or by the computationally much cheaper frugal coarse space, which numerically proved to be robust for many realistic coefficient distributions. In this article, both, the adaptive and the frugal FETI-DP methods are applied to discretizations obtained by using virtual elements. As model problems, stationary diffusion and compressible linear elasticity in two spatial dimensions are considered. The performance of the methods is numerically tested, varying the quasi-uniformity of the underlying meshes, the polynomial degree, the scaling method, and considering regular and irregular domain decompositions. It is shown that adaptive and frugal FETI-DP for virtual elements behave similarly as in the finite element case.

Adaptive Nonlinear Domain Decomposition Methods with an Application to the $p$-Laplacian

In this article, different nonlinear domain decomposition methods are applied to nonlinear problems with highly heterogeneous coefficient functions with jumps. In order to obtain a robust solver with respect to nonlinear as well as linear convergence, adaptive coarse spaces are employed. First, as an example for a nonlinearly left-preconditioned domain decomposition method, the two-level restricted nonlinear Schwarz method H1-RASPEN (Hybrid Restricted Additive Schwarz Preconditioned Exact Newton) is combined with an adaptive GDSW (generalized Dryja--Smith--Widlund) coarse space. Second, as an example for a nonlinearly right-preconditioned domain decomposition method, a nonlinear FETI-DP (Finite Element Tearing and Interconnecting--Dual Primal) method is equipped with an edge-based adaptive coarse space. Both approaches are compared with the respective nonlinear domain decomposition methods with classical coarse spaces as well as with the respective Newton--Krylov methods with adaptive coarse spaces. For some two-dimensional $p$-Laplace model problems with different spatial coefficient distributions, it can be observed that the best linear and nonlinear convergence can only be obtained when combining the nonlinear domain decomposition methods with adaptive coarse spaces.

A frugal FETI-DP and BDDC coarse space for heterogeneous problems

The convergence rate of domain decomposition methods is generally determined by the eigenvalues of the preconditioned system. For second-order elliptic partial differential equations, coefficient discontinuities with a large contrast can lead to a deterioration of the convergence rate. Only by implementing an appropriate coarse space, or second level, a robust domain decomposition method can be obtained. In this article, a new frugal coarse space for FETI-DP (Finite Element Tearing and Interconnecting-Dual Primal) and BDDC (Balancing Domain Decomposition by Constraints) methods is presented, which has a lower set-up cost than competing adaptive coarse spaces. In particular, in contrast to adaptive coarse spaces, it does not require the solution of any local generalized eigenvalue problems. The approach considered here aims at a low-dimensional approximation of the adaptive coarse space by using appropriate weighted averages, and it is robust for a broad range of coefficient distributions for diffusion and elasticity problems. However, in general, for completely arbitrary coefficient distributions with high contrast, some additional, adaptively chosen constraints are necessary in order to guarantee robustness. In this article, the robustness is heuristically justified as well as numerically shown for several coefficient distributions. The new coarse space is compared to adaptive coarse spaces, and parallel scalability up to 262144 parallel cores for a parallel BDDC implementation with the new coarse space is shown. The superiority of the new coarse space over classic coarse spaces with respect to parallel weak scalability and time-to-solution is confirmed by numerical experiments. Since the new frugal coarse space is computationally inexpensive, it could serve as a new default coarse space, which, for very challenging coefficient distributions, could then still be enhanced by adaptively chosen constraints.

FETI-DP for the Three Dimensional Virtual Element Method

We deal with the Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) preconditioner for elliptic problems discretized by the virtual element method (VEM). We extend the result of [22] to the three dimensional case. We prove polylogarithmic condition number bounds, independent of the number of subdomains, the mesh size, and jumps in the diffusion coefficients. Numerical experiments validate the theory

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 for the virtual element method

We build and analyze Balancing Domain Decomposition by Constraint (BDDC) and Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) preconditioners for elliptic problems discretized by the virtual element method (VEM). We prove polylogarithmic condition number bounds, independent of the number of subdomains, the mesh size, and jumps in the diffusion coefficients. Numerical experiments confirm the theory.

Nonlinear FETI-DP and BDDC Methods: A Unified Framework and Parallel Results

Parallel Newton--Krylov FETI-DP (Finite Element Tearing and Interconnecting---Dual-Primal) domain decomposition methods are fast and robust solvers, e.g., for nonlinear implicit problems in structural mechanics. In these methods, the nonlinear problem is first linearized and then decomposed into loosely coupled (linear) problems, which can be solved in parallel. By changing the order of the operations, new parallel communication can be constructed, where the loosely coupled local problems are nonlinear. We discuss different nonlinear FETI-DP methods which are equivalent when applied to linear problems but which show a different performance for nonlinear problems. Moreover, a new unified framework is introduced which casts all nonlinear FETI-DP domain decomposition approaches discussed in the literature into a single algorithm. Furthermore, the equivalence of nonlinear FETI-DP methods to specific nonlinearly right-preconditioned Newton--Krylov methods is shown. For the methods using nested Newton iterations, a strategy is presented to stop the inner Newton iteration early, resulting in an approximate local nonlinear elimination. Additionally, the nonlinear BDDC (Balancing Domain Decomposition by Constraint) method is presented as a right-preconditioned Newton approach. Finally, for the first time, parallel weak scaling results for four different nonlinear FETI-DP approaches are compared to standard Newton--Krylov FETI-DP in two and three dimensions, using both exact as well as highly scalable inexact linear FETI-DP preconditioners and up to 131,072 message passing interface (MPI) ranks on the JUQUEEN supercomputer at Forschungszentrum Jülich. For a model problem with nonlocal nonlinearities, nonlinear FETI-DP methods are shown to be up to five times faster than the standard Newton--Krylov FETI-DP approach.

Toward Extremely Scalable Nonlinear Domain Decomposition Methods for Elliptic Partial Differential Equations

The solution of nonlinear problems, e.g., in material science, requires fast and highly scalable parallel solvers. Finite element tearing and interconnecting dual primal (FETI-DP) domain decomposition methods are parallel solution methods for implicit problems discretized by finite elements. Recently, nonlinear versions of the well-known FETI-DP methods for linear problems have been introduced. In these methods, the nonlinear problem is decomposed before linearization. This approach can be viewed as a strategy to further localize computational work and to extend the parallel scalability of FETI-DP methods toward extreme-scale supercomputers. Here, a recent nonlinear FETI-DP method is combined with an approach that allows an inexact solution of the FETI-DP coarse problem. We combine the nonlinear FETI-DP domain decomposition method with an algebraic multigrid (AMG) method and thus obtain a hybrid nonlinear domain decomposition/multigrid method. We consider scalar nonlinear problems as well as nonlinear hyp...

Parallel simulation of patient‐specific atherosclerotic arteries for the enhancement of intravascular ultrasound diagnostics

PurposeThe purpose of this paper is to present a computational framework for the simulation of patient‐specific atherosclerotic arterial walls. Such simulations provide information regarding the mechanical stress distribution inside the arterial wall and may therefore enable improved medical indications for or against medical treatment. In detail, the paper aims to provide a framework which takes into account patient‐specific geometric models obtained by in vivo measurements, as well as a fast solution strategy, giving realistic numerical results obtained in reasonable time.Design/methodology/approachA method is proposed for the construction of three‐dimensional geometrical models of atherosclerotic arteries based on intravascular ultrasound virtual histology data combined with angiographic X‐ray images, which are obtained on a routine basis in the diagnostics and medical treatment of cardiovascular diseases. These models serve as a basis for finite element simulations where a large number of unknowns need to be calculated in reasonable time. Therefore, the finite element tearing and interconnecting‐dual primal (FETI‐DP) domain decomposition method is applied, to achieve an efficient parallel solution strategy.FindingsIt is shown that three‐dimensional models of patient‐specific atherosclerotic arteries can be constructed from intravascular ultrasound virtual histology data. Furthermore, the application of the FETI‐DP domain decomposition method leads to a fast numerical framework. In a numerical example, the importance of three‐dimensional models and thereby fast solution algorithms is illustrated by showing that two‐dimensional approximations differ significantly from the 3D solution.Originality/valueThe decision for or against intravascular medical treatment of atherosclerotic arteries strongly depends on the mechanical situation of the arterial wall. The framework presented in this paper provides computer simulations of stress distributions, which therefore enable improved indications for medical methods of treatment.

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.

BDDC and FETI-DP without matrices or vectors

The connection between the BDDC (balancing domain decomposition by constraints) algorithm and the FETI-DP (finite element tearing and interconnecting dual–primal) algorithm is discussed in the language of function spaces, operators and linear functionals.

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.