Fast estimation of discretization error for FE problems solved by domain decomposition

This paper presents a strategy for a posteriori error estimation for substructured problems solved by non-overlapping domain decomposition methods. We focus on global estimates of the discretization error obtained through the error in constitutive relation for linear mechanical problems. Our method allows to compute error estimate in a fully parallel way for both primal (BDD) and dual (FETI) approaches of non-overlapping domain decomposition whatever the state (converged or not) of the associated iterative solver. Results obtained on an academic problem show that the strategy we propose is efficient in the sense that correct estimation is obtained with fully parallel computations; they also indicate that the estimation of the discretization error reaches sufficient precision in very few iterations of the domain decomposition solver, which enables to consider highly effective adaptive computational strategies.     

Robust FETI-DP methods for heterogeneous three dimensional elasticity problems

Iterative substructuring methods with Lagrange multipliers are considered for heterogeneous linear elasticity problems with large discontinuities in the material stiffnesses. In particular, results for algorithms belonging to the family of dual-primal FETI methods are presented. The core issue of these algorithms is the construction of an appropriate global problem, in order to obtain a robust method which converges independently of the material discontinuities. In this article, several necessary and sufficient conditions arising from the theory are numerically tested and confirmed. Furthermore, results of numerical experiments are presented for situations which are not covered by the theory, such as curved edges and material discontinuities not aligned with the interface, and an attempt is made to develop rules for these cases.     

A new era in scientific computing: Domain decomposition methods in hybrid CPU–GPU architectures

Recent advances in graphics processing units (GPUs) technology open a new era in high performance computing. Applications of GPUs to scientific computations are attracting a lot of attention due to their low cost in conjunction with their inherently remarkable performance features and the recently enhanced computational precision and improved programming tools. Domain decomposition methods (DDM) constitute today an important category of methods for the solution of highly demanding problems in simulation-based applied science and engineering. Among them, dual domain decomposition methods have been successfully applied in a variety of problems in both sequential as well as in parallel/distributed processing systems. In this work, we demonstrate the implementation of the FETI method to a hybrid CPU–GPU computing environment. Parametric tests on implicit finite element structural mechanics benchmark problems revealed the tremendous potential of this type of hybrid computing environment as a result of the full exploitation of multi-core CPU hardware resources and the intrinsic software and hardware features of the GPUs as well as the numerical properties of the solution method.     

Total FETI based algorithm for contact problems with additional non-linearities

The paper is concerned with application of a new variant of the FETI domain decomposition method called Total FETI to the solution to contact problems. Its basic idea is that both the compatibility between adjacent sub-domains and Dirichlet boundary conditions are enforced by Lagrange multipliers. We introduce the Total FETI technique for solution to the variational inequalities governing the equilibrium of system of bodies in contact. Moreover, we show implementation of the method into a code which treats the material and geometric non-linear effects. Numerical experiments were carried out with our in-house general purpose finite element package PMD.     

Iterative methods for domain decomposition using spectral tau and collocation approximation

In this paper we consider the numerical approximation of elliptic problems by spectral methods in domains subdivided into substructures. We review an iterative procedure with interface relaxation, reducing the given differential problem to a sequence of Dirichlet and mixed Neumann-Dirichlet problems on each subdomain. The iterative procedure is applied to both tau and collocation spectral approximations. Two optimal strategies for the automatic selection of the relaxation parameter are indicated. We present several numerical experiments showing the convergence properties of the iterative scheme with respect to the decomposition. A multilevel technique for domain decomposition methods is proposed.     

A numerical scheme to couple subdomains with different time-steps for predominantly linear transient analysis

This paper generalizes the explicit/implicit time-integration algorithms pioneered by Belytschko, Hughes and their respective co-workers, and the FETI domain decomposition methods introduced by Farhat and his co-workers, to the case where the same Newmark scheme, but different β and γ coefficients and different time-steps, are specified in each subdomain. Building upon the work of Farhat, Crivelli and Géradin, it considers various interface boundary constraints and performs a stability analysis of the proposed time integration algorithms.     

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 Jülich show the potential of these methods. A hyperelastic problem from biomechanics is presented as an application of the methods to nonlinear finite element analysis.     

Schwarz iterations for the efficient solution of screen problems with boundary elements

This paper investigates two domain decomposition algorithms for the numerical solution of boundary integral equations of the first kind. The schemes are based on theh-type boundary element Galerkin method to which the multiplicative and the additive Schwarz methods are applied. As for twodimensional problems, the rates of convergence of both methods are shown to be independent of the number of unknowns. Numerical results for standard model problems arising from Laplaces' equation with Dirichlet or Neumann boundary conditions in both two and three dimensions are discussed. A multidomain decomposition strategy is indicated by means of a screen problem in three dimensions, so as to obtain satisfactory experimental convergence rates.     

Uzawa Conjugate Gradient Domain Decomposition Methods for Coupled Stokes Flows

This paper deals with nonoverlapping domain decomposition methods for two coupled Stokes flows, based on the duality theory. By introducing a fictitious variable in the transmission condition and using saddle-point equations, the problem is restated as a linearly constrained maximization problem. According to whether constraints are uncoupled Stokes problems or uncoupled Poisson problems, two Uzawa-type domain decomposition algorithms are proposed. The results of some numerical experiments on a model problem are given.     

Aerodynamics of high speed trains passing by each other

A three-dimensional flow field induced by two trains passing by each other inside a tunnel is studied based on the numerical simulation of the three-dimensional compressible Euler/Navier-Stokes equations formulated in the finite difference approximation. A domain decomposition method with the FSA (fortified solution algorithm) interface scheme is used to treat this moving-body problem. The computed results show the basic characteristics of the flow field created when two trains pass by each other. The history of the pressure distributions and the aerodynamic forces acting on the trains are the main areas discussed. The results indicate that the phenomenon is complicated due to the interaction of the flow induced by the two trains. Strong side forces occur between the two trains when the front portion of the opposite train passes by. The forces fluctuate rapidly and the maximum suction force occurs when two trains are aligned side by side. The results also indicate the effectiveness of the present numerical method calculating moving boundary problems.     

A monolithic energy conserving method to couple heterogeneous time integrators with incompatible time steps in structural dynamics

A new hybrid multi-time method for multi-time scales structural dynamics simulations is described. A monolithic method in a Schur dual domain decomposition framework is proposed and allows to consider heterogeneous time integrators with their own time discretization and possible large ratio between the time steps for each subdomain. In the proposed method, zero numerical dissipation is ensured at the interface. This implies that the global stability of the coupling method is governed by the stability of each time integrator without influence of the interface. For that purpose, velocity continuity is ensured in a weak sense at the interfaces, and time integrators (Newmark, HHT, Simo, Krenk, Verlet) are introduced in a unified framework (incremental velocity formulation). Furthermore, dynamics governing equations are introduced from a weak formulation in time. In other words, equilibrium equation is no more ensured in a strong sense at a given time step, but rather on average on a time interval. Some numerical examples illustrate the efficiency and the robustness of the proposed method, for ratio of time scales close to 1000 without any numerical dissipation at the interfaces.     

An augmented Lagrangian decomposition method for quasi-separable problems in MDO

Several decomposition methods have been proposed for the distributed optimal design of quasi-separable problems encountered in Multidisciplinary Design Optimization (MDO). Some of these methods are known to have numerical convergence difficulties that can be explained theoretically. We propose a new decomposition algorithm for quasi-separable MDO problems. In particular, we propose a decomposed problem formulation based on the augmented Lagrangian penalty function and the block coordinate descent algorithm. The proposed solution algorithm consists of inner and outer loops. In the outer loop, the augmented Lagrangian penalty parameters are updated. In the inner loop, our method alternates between solving an optimization master problem and solving disciplinary optimization subproblems. The coordinating master problem can be solved analytically; the disciplinary subproblems can be solved using commonly available gradient-based optimization algorithms. The augmented Lagrangian decomposition method is derived such that existing proofs can be used to show convergence of the decomposition algorithm to Karush–Kuhn–Tucker points of the original problem under mild assumptions. We investigate the numerical performance of the proposed method on two example problems.     

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.     

Improved interface conditions for 2D domain decomposition with corners: a theoretical determination

This article deals with a local improvement of domain decomposition methods for 2-dimensional elliptic problems for which either the geometry or the domain decomposition presents conical singularities. The problem amounts to determining the coefficients of interface boundary conditions so that the domain decomposition algorithm has rapid convergence. Specific problems occur in the presence of conical singularities. Starting from the method used for regular interfaces, we derive a local improvement by matching the singularities, that is, the initial terms of the asymptotic expansion arond the corner, provided by Kondratiev theory. This theoretical approach leads to the explicit computation of coefficients in the interface boundary conditions, which have been tested numerically. This final numerical step is presented in a companion article [4]. This article focuses on the method used to compute these coefficients and provides detailed examples for a model problem. C. Chniti was supported by the Tunisian Government and the école Polytechnique and visited IRMAR-Rennesl as a post-doctoral assistant.     

A preconditioner for the Schur complement matrix

A preconditioner for iterative solution of the interface problem in Schur Complement Domain Decomposition Methods is presented. This preconditioner is based on solving a global problem in a narrow strip around the interface. It requires much less memory and computing time than classical Neumann–Neumann preconditioner and its variants, and handles correctly the flux splitting among subdomains that share the interface. The aim of this work is to present a theoretical basis (regarding the behavior of Schur complement matrix spectra) and some simple numerical experiments conducted in a sequential environment as a motivation for adopting the proposed preconditioner. Efficiency, scalability, and implementation details on a production parallel finite element code [Sonzogni V, Yommi A, Nigro N, Storti M. A parallel finite element program on a Beowulf cluster. Adv Eng Software 2002;33(7–10):427–43; Storti M, Nigro N, Paz R, Dalcín L. PETSc-FEM: a general purpose, parallel, multi-physics FEM program, 1999–2006] can be found in works [Paz R, Storti M. An interface strip preconditioner for domain decomposition methods: application to hydrology. Int J Numer Methods Eng 2005;62(13):1873–94; Paz R, Nigro N, Storti M. On the efficiency and quality of numerical solutions in cfd problems using the interface strip preconditioner for domain decomposition methods. Int J Numer Methods Fluids, in press].     

Parallel solution of contact shape optimization problems based on Total FETI domain decomposition method

An application of a variant of the parallel domain decomposition method that we call Total FETI or TFETI (Total Finite Element Tearing and Interconnecting) for the solution of contact problems of elasticity to the parallel solution of contact shape optimization problems is described. A unique feature of the TFETI algorithm is its capability to solve large contact problems with optimal, i.e., asymptotically linear complexity. We show that the algorithm is even more efficient for the solution of the contact shape optimization problems as it can exploit effectively a specific structure of the auxiliary problems arising in the semi-analytic sensitivity analysis. Thus the triangular factorizations of the stiffness matrices of the subdomains are carried out in parallel only once for each design step, the evaluation of the components of the gradient of the cost function can be carried out in parallel, and even the evaluation of each component of the gradient itself can be further parallelized using the standard TFETI scheme. Theoretical results which prove asymptotically linear complexity of the solution are reported and documented by numerical experiments. The results of numerical solution of a 3D contact shape optimization problem confirm the high degree of parallelism of the algorithm.     

Coupling projection domain decomposition method and Kansa's method in electrostatic problems

This paper present a novel approach for solving electrostatic problems associated with an asymmetrical shielded stripline and shielded coupled-striplines. This novel approach is based on combination of radial basis functions-based meshless unsymmetric collocation method (also, Kansa's method) with projection domain decomposition method. Under this new method, we just need to solve a Steklov-Poincaré interface equation and the original problem is solved by computing a series of independent subproblems. Two real problems are solved by the proposed approach to demonstrate the accuracy and efficiency.     

Goal oriented error estimators for Stokes equations based on variationally consistent postprocessing

In many flow problems, and in particular when fluid-structure interaction is considered, the important unknowns are the forces acting on the structure in certain areas. Hence, accurate values for these local quantities is essentially what one wants to get out of the flow computations. By means of variationally consistent postprocessing, where forces are computed using the weak form of the equations, we recover the requested forces. Goal oriented local error indicators are provided by solving an auxiliary problem. At the end numerical examples are presented that illustrate how this goal oriented strategy gives improved efficiency compared to traditional methods. The fluid flow is assumed to be governed by the Stokes equations.     

Optimized Schwarz method without overlap for the gravitational potential equation on cluster of graphics processing unit

Many engineering and scientific problems need to solve boundary value problems for partial differential equations or systems of them. For most cases, to obtain the solution with desired precision and in acceptable time, the only practical way is to harness the power of parallel processing. In this paper, we present some effective applications of parallel processing based on hybrid CPU/GPU domain decomposition method. Within the family of domain decomposition methods, the so-called optimized Schwarz methods have proven to have good convergence behaviour compared to classical Schwarz methods. The price for this feature is the need to transfer more physical information between subdomain interfaces. For solving large systems of linear algebraic equations resulting from the finite element discretization of the subproblem for each subdomain, Krylov method is often a good choice. Since the overall efficiency of such methods depends on effective calculation of sparse matrix–vector product, approaches that use graphics processing unit (GPU) instead of central processing unit (CPU) for such task look very promising. In this paper, we discuss effective implementation of algebraic operations for iterative Krylov methods on GPU. In order to ensure good performance for the non-overlapping Schwarz method, we propose to use optimized conditions obtained by a stochastic technique based on the covariance matrix adaptation evolution strategy. The performance, robustness, and accuracy of the proposed approach are demonstrated for the solution of the gravitational potential equation for the data acquired from the geological survey of Chicxulub crater.     

Domain decomposition algorithms for spectral methods

In this paper we review some multiple-domain decomposition algorithrns for spectral methods. The alternating Schwarz algorithm and a flux preserving domain decomposition algorithm are considered. For both of them, some theoretical convergence results are given. These domain decomposition algorithms allow the use of iterative procedures for block matrices that can be efficiently implemented on parallel processors. Several numerical experiences based on the Chebyshev collocation method are carried out. A comparison of the performances of the two algorithms on several test problems is presented.