Abstract

In this work, a domain decomposition strategy for non-linear hyper-reduced-order models is presented. The basic idea consists of restricting the reduced-order basis functions to the nodes belonging to each of the subdomains into which the physical domain is partitioned. An extension of the proposed domain decomposition strategy to a hybrid full-order/reduced-order model is then described. The general domain decomposition approach is particularized for the reduced-order finite element approximation of the incompressible Navier–Stokes equations with hyper-reduction. When solving the reduced incompressible Navier–Stokes equations, instabilities in the form of large gradients of the recovered reduced-order unknown at the subdomain interfaces may appear, which is the motivation for the design of additional stability terms giving rise to penalty matrices. Numerical examples illustrate the behavior of the proposed method for the simulation of the reduced-order systems, showing the capability of the approach to adapt to configurations which are not present in the original snapshot set.

Abstract

Creating a highly parallelizable code is a challenge specially for Distributed Memory Machines (DMMs). Moreover, algorithms and data structures suitable for these platforms can be very different from the ones used in serial code. For this reason, many programmers in the field prefer to start their own code from scratch. However, for an already existing framework supported by a long-time expertise the idea of transformation becomes attractive in order to reuse the effort done during years of development. In this presentation we explain how a relatively complex framework but with modular structure can be prepared for high performance computing with minimum modification. Kratos Multi-Physics is an open source generic multi-disciplinary platform for solution of coupled problems consist of fluid, structure, thermal and electromagnetic fields. The parallelization of this framework is performed with objective of enforcing the less possible changes to its different solver modules and encapsulate the changes as much as possible in its common kernel. This objective is achieved thanks to the Kratos design and also innovative way of dealing with data transfers for a multi-disciplinary code. This work is completed by the migration of the framework from the 86× architecture to the Marenostrum Supercomputing platform. The migration has been verified by a set of benchmarks which show high scalability, from which we present the Telescope problem in this paper.

Abstract

In this paper, we analyse the numerical approximation of the heat transfer problem between two subdomains that we will consider filled with a fluid and separated by a thin solid wall. First of all, we state the problem in the whole domain with discontinuous physical properties. As an alternative and under certain assumptions on the separating walls, a classical Robin boundary condition between the fluid domains is obtained, thus eliminating the solid wall, and according to which the heat flux is proportional to the temperature difference between the two subdomains. Apart from discussing the relation between both approaches, we consider their numerical approximation, considering different alternatives for the first case, that is, the case in which temperatures are also computed in the solid wall.

Abstract

In this work, we propose an enhanced implementation of balancing Neumann-Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticity problems. On one hand, we propose a novel way to deal with singular matrices and pseudo-inverses appearing in local solvers. It is based on a kernel identication strategy that allows us to eciently compute the action of the pseudo-inverse via local indenite solvers. We further show how, identifying a minimum set of degrees of freedom to be xed, an equivalent denite system can be solved instead, even in the elastic case. On the other hand, we propose a simple modication of the preconditioned conjugate gradient (PCG) algorithm that reduces the number of Dirichlet solvers to only one per iteration, leading to similar computational cost as additive methods. After these improvements of the BNN PCG algorithm, we compare its performance against that of the BDDC preconditioners on a pair of large-scale distributed-memory platforms. The enhanced BNN method is a competitive preconditioner for three-dimensional Poisson and elasticity problems, and outperforms the BDDC method in many cases.

Abstract

In this work we propose a novel parallelization approach of two-level balancing domain decomposition by constraints preconditioning based on overlapping of fine-grid and coarse-grid duties in time. The global set of MPI tasks is split into those that have fine-grid duties and those that have coarse-grid duties, and the different computations and communications in the algorithm are then re-scheduled and mapped in such a way that the maximum degree of overlapping is achieved while preserving data dependencies among them. In many ranges of interest, the extra cost associated to the coarse-grid problem can be fully masked by fine-grid related computations (which are embarrassingly parallel). Apart from discussing code implementation details, the paper also presents a comprehensive set of numerical experiments, that includes weak scalability analyses, with structured and unstructured meshes, and exact and inexact solvers for the 3D Poisson and linear elasticity problems on a pair of state-of-the-art multicore-based distributed-memory machines. This experimental study reveals remarkable weak scalability in the solution of problems with thousands of millions of unknowns on several tens of thousands of computational cores.

Abstract

In this work, we analyze the scalability of inexact two-level balancing domain decomposition by constraints (BDDC) preconditioners for Krylov subspace iterative solvers, when using a highly scalable asynchronous parallel implementation where fine and coarse correction computations are overlapped in time. This way, the coarse-grid problem can be fully overlapped by fine-grid computations (which are embarrassingly parallel) in a wide range of cases. Further, we consider inexact solvers to reduce the computational cost/complexity and memory consumption of coarse and local problems and boost the scalability of the solver. Out of our numerical experimentation, we conclude that the BDDC preconditioner is quite insensitive to inexact solvers. In particular, one cycle of algebraic multigrid (AMG) is enough to attain algorithmic scalability. Further, the clear reduction of computing time and memory requirements of inexact solvers compared to sparse direct ones makes possible to scale far beyond state-of-the-art BDDC implementations. Excellent weak scalability results have been obtained with the proposed inexact/overlapped implementation of the two-level BDDC preconditioner, up to 93,312 cores and 20 billion unknowns on JUQUEEN. Further, we have also applied the proposed setting to unstructured meshes and partitions for the pressure Poisson solver in the backward-facing step benchmark domain.

Abstract

In this paper we present a fully distributed, communicator-aware, recursive, and interlevel-overlapped message-passing implementation of the multilevel balancing domain decomposition by constraints (MLBDDC) preconditioner. The implementation highly relies on subcommunicators in order to achieve the desired effect of coarse-grain overlapping of computation and communication, and communication and communication among levels in the hierarchy (namely, interlevel overlapping). Essentially, the main communicator is split into as many nonoverlapping subsets of message-passing interface (MPI) tasks (i.e., MPI subcommunicators) as levels in the hierarchy. Provided that specialized resources (cores and memory) are devoted to each level, a careful rescheduling and mapping of all the computations and communications in the algorithm lets a high degree of overlapping be exploited among levels. All subroutines and associated data structures are expressed recursively, and therefore MLBDDC preconditioners with an arbitrary number of levels can be built while re-using significant and recurrent parts of the codes. This approach leads to excellent weak scalability results as soon as level-1 tasks can fully overlap coarser-levels duties. We provide a model to indicate how to choose the number of levels and coarsening ratios between consecutive levels and determine qualitatively the scalability limits for a given choice. We have carried out a comprehensive weak scalability analysis of the proposed implementation for the three-dimensional Laplacian and linear elasticity problems on structured and unstructured meshes. Excellent weak scalability results have been obtained up to 458,752 IBM BG/Q cores and 1.8 million MPI being, being the first time that exact domain decomposition preconditioners (only based on sparse direct solvers) reach these scales. (An erratum is attached.)

Abstract

Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in‐plane–out‐of‐plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user‐defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution.

Abstract

The NURBS-enhanced finite element method (NEFEM) combined with a hybridisable discontinuous Galerkin (HDG) approach is presented for the first time. The proposed technique completely eliminates the uncertainty induced by a polynomial approximation of curved boundaries that is common within an isoparametric approach and, compared to other DG methods, provides a significant reduction in number of degrees of freedom. In addition, by exploiting the ability of HDG to compute a postprocessed solution and by using a local a priori error estimate valid for elliptic problems, an inexpensive, reliable and computable error estimator is devised. The proposed methodology is used to solve Stokes flow problems using automatic degree adaptation. Particular attention is paid to the importance of an accurate boundary representation when changing the degree of approximation in curved elements. Several strategies are compared and the superiority and reliability of HDG-NEFEM with degree adaptation is illustrated.

Abstract

Two parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are analyzed and numerically studied for implicit time discretizations of the Bidomain equations. This system models 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 non-linear algebraic system arises from a finite element discretization in space and an implicit discretization in time, based on decoupling the PDEs from the ODEs. Within each Newton iteration, the Jacobian linear system is solved by a Krylov method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the recently introduced deluxe scaling of the dual variables. Several parallel numerical tests on Linux clusters confirm a novel polylogarithmic convergence rate bound, showing scalability and quasi-optimality of the proposed solvers.