Nonlinear FETI-DP and BDDC Methods

In the simulation of deformation processes in material science the consideration of a microscopic material structure is often necessary, as in the simulation of modern high strength steels. A straightforward finite element discretization of the complete deformed body resolving the microscopic structure leads to very large nonlinear problems and a solution is out of reach, even on modern supercomputers. In homogenization approaches, as the computational scale bridging approach FE2, the macroscopic scale of the deformed object is decoupled from the microscopic scale of the material structure. These approaches only consider the microstructure in a localized fashion on independent and parallel representative volume elements (RVEs). This introduces massive parallelism on the macroscopic level and is thus ideal for modern computer architectures with large numbers of parallel computational cores. Nevertheless, the discretization of an RVE can still result in large nonlinear problems and thus highly scalable parallel solvers are necessary. In this context, nonlinear FETI-DP (Finite Element Tearing and Interconnecting - Dual-Primal) and BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods are discussed in this thesis, which are parallel solution methods for nonlinear problems arising from a finite element discretization. These approaches can be viewed as a strategies to further localize the computational work and to extend the parallel scalability of classical FETI-DP and BDDC methods towards extreme-scale supercomputers. Also variants providing an inexact solution of the FETI-DP coarse problem are considered in this thesis, combining two successful paradigms, i.e., nonlinear domain decomposition and AMG (Algebraic Multigrid). An efficient implementation of the resulting inexact reduced Nonlinear-FETI-DP-1 method is presented and scalability beyond 200,000 computational cores is showed. Finally, a highly scalable FE2 implementation using recent inexact reduced FETI-DP methods to solve the RVE problems on the microscopic level is presented and scalability on all 458,752 cores of the JUQUEEN BlueGene/Q system at Forschungszentrum Jülich is demonstrated

BDDC and FETI-DP under Minimalist Assumptions

The FETI-DP, BDDC and P-FETI-DP preconditioners are derived in a particulary simple abstract form. It is shown that their properties can be obtained from only on a very small set of algebraic assumptions. The presentation is purely algebraic and it does not use any particular definition of method components, such as substructures and coarse degrees of freedom. It is then shown that P-FETI-DP and BDDC are in fact the same. The FETI-DP and the BDDC preconditioned operators are of the same algebraic form, and the standard condition number bound carries over to arbitrary abstract operators of this form. The equality of eigenvalues of BDDC and FETI-DP also holds in the minimalist abstract setting. The abstract framework is explained on a standard substructuring example.Comment: 11 pages, 1 figure, also available at http://www-math.cudenver.edu/ccm/reports

High-Performance Computing Two-Scale Finite Element Simulations of a Contact Problem Using Computational Homogenization - Virtual Forming Limit Curves for Dual-Phase Steel

The appreciated macroscopic properties of dual-phase (DP) steels strongly depend on their microstructure. Therefore, accurate finite element (FE) simulations of a deformation process of such a steel require the incorporation of the microscopic heterogeneous structure. Usually, a brute force FE discretization incorporating the microstructure is not feasible since it results in exceedingly large problem sizes. Instead, the microstructure has to be incorporated by using computational homogenization. We present a numerical two-scale approach of the Nakajima test for a DP steel, which is a well known material test in the steel industry. It can be used to derive forming limit diagrams (FLDs), which allow experts to judge the maximum formability properties of a specific type of sheet metal in the considered thickness. For the simulations, we use our software package FE2TI, which is a highly scalable implementation of the well known FE2 homogenization approach. The microstructure is represented by a representative volume element (RVE) and it is discretized separately from the macroscopic problem. We discuss the incorporation of contact constraints using a penalty formulation as well as appropriate boundary conditions. In addition, we introduce a simple load step strategy and different opportunities for the choice of an initial value for a single load step by using an interpolation polynomial. Finally, we come up with computationally derived FLDs. Although we use a computational homogenization strategy, the resulting problems on both scales can be quite large. The efficient solution of such large problems requires parallel strategies. Therefore, we consider the highly scalable nonlinear domain decomposition methods FETI-DP (Finite Element Tearing and Interconnecting - Dual-Primal) and BDDC (Balancing Domain Decomposition by Constraints). For the first time, the BDDC approach is used for the parallel solution of the macroscopic problem in a simulation of the Nakajima test. We introduce a unified framework that combines all variants of nonlinear FETI-DP and nonlinear BDDC. For the first time, we introduce a nonlinear FETI-DP variant that chooses suitable elimination sets by utilizing information from the nonlinear residual. Furthermore, we show weak scaling results for different nonlinear FETI-DP variants and several model problems

Balancing domain decomposition by constraints and perturbation

In this paper, we formulate and analyze a perturbed formulation of the balancing domain decomposition by constraints (BDDC) method. We prove that the perturbed BDDC has the same polylogarithmic bound for the condition number as the standard formulation. Two types of properly scaled zero-order perturbations are considered: one uses a mass matrix, and the other uses a Robin-type boundary condition, i.e, a mass matrix on the interface. With perturbation, the wellposedness of the local Neumann problems and the global coarse problem is automatically guaranteed, and coarse degrees of freedom can be defined only for convergence purposes but not well-posedness. This allows a much simpler implementation as no complicated corner selection algorithm is needed. Minimal coarse spaces using only face or edge constraints can also be considered. They are very useful in extreme scale calculations where the coarse problem is usually the bottleneck that can jeopardize scalability. The perturbation also adds extra robustness as the perturbed formulation works even when the constraints fail to eliminate a small number of subdomain rigid body modes from the standard BDDC space. This is extremely important when solving problems on unstructured meshes partitioned by automatic graph partitioners since arbitrary disconnected subdomains are possible. Numerical results are provided to support the theoretical findings.Peer ReviewedPostprint (published version

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

Adaptive Three-level BDDC Using Frugal Constraints

The highly parallel scalable three-level BDDC (Balancing Domain Decomposition by Constraints) method is a very successful approach to overcome the scaling bottleneck of directly solving a large coarse problem in classical two-level BDDC. As long as the problem is homogeneous on each subregion, three-level BDDC is also provably robust in many cases. For problems with complex microstructures, as, e.g., stationary diffusion problems with jumps in the diffusion coefficient function, in two-level BDDC methods, advanced adaptive or frugal coarse spaces have to be considered to obtain a robust preconditioner. Unfortunately, these approaches result in even larger coarse problems on the second level and, additionally, computing adaptive coarse constraints is computationally expensive. Therefore, in this article, the three-level approach is combined with a provably robust adaptive coarse space and the computationally cheaper frugal coarse space. Both coarse spaces are used on the second as well as the third level. All different possible combinations are investigated numerically for stationary diffusion problems with highly varying coefficient functions