FETI Method Research Articles

Two-level adaptation for Adaptive Multipreconditioned FETI

This article introduces two strategies to reduce the memory cost of the Adaptive Multipreconditioned FETI method (AMPFETI) while preserving its capability to solve ill conditioned systems efficiently. Their common principle is to gather search directions into aggregates which are frequently adapted in order to achieve the best compromise between the decrease of the solver error and the computational resources employed. The methods are assessed on two weak scalability studies on highly heterogeneous problems up to 10,368 cores and half a billion of unknowns, and on two ill-conditioned industrial applications, related to the numerical homogenization of solid propellant and to the simulation of a multiperforated aircraft combustion chamber.

Recycling of solution spaces in multipreconditioned FETI methods applied to structural dynamics

SummaryThis article presents a new method to recycle the solution space of an adaptive multipreconditioned finite element tearing and interconnecting algorithm in the case where the same operator is solved for multiple right‐hand sides like in linear structural dynamics. It accelerates the computation from the second time step on by applying a coarse space that is generated from Ritz approximations of local eigenproblems, using the solution space of the first time step. These eigenproblems are known to provide very efficient coarse spaces but must usually be solved a priori at high computational cost. Their Ritz approximations are much smaller and less expensive to solve. Recycling methods based on Ritz approximations of global eigenproblems have been published for classical finite element tearing and interconnecting algorithms, but their efficient application to multipreconditioned variants is not possible. This article also presents the application of a simpler recycling procedure, which reuses plain solution spaces, to adaptive multipreconditioned finite element tearing and interconnecting. Numerical results of the application of the presented methods to four test cases are shown. The new Ritz approximation method leads to coarse spaces, which turn out to be as efficient as those obtained from solving the unreduced eigenproblems. It is the most efficient recycling method currently available for multipreconditioned dual domain decomposition techniques.

A mixed parallel strategy for the solution of coupled multi-scale problems at finite strains

A mixed parallel strategy for the solution of homogenization-based multi-scale constitutive problems undergoing finite strains is proposed. The approach aims to reduce the computational time and memory requirements of non-linear coupled simulations that use finite element discretization at both scales (FE $$^2$$ ). In the first level of the algorithm, a non-conforming domain decomposition technique, based on the FETI method combined with a mortar discretization at the interface of macroscopic subdomains, is employed. A master–slave scheme, which distributes tasks by macroscopic element and adopts dynamic scheduling, is then used for each macroscopic subdomain composing the second level of the algorithm. This strategy allows the parallelization of FE $$^2$$ simulations in computers with either shared memory or distributed memory architectures. The proposed strategy preserves the quadratic rates of asymptotic convergence that characterize the Newton–Raphson scheme. Several examples are presented to demonstrate the robustness and efficiency of the proposed parallel strategy.

The impact of enabling multiple subdomains per MPI process in the TFETI domain decomposition method

The paper deals with handling multiple subdomains per computational core in the PERMON toolbox, namely in the PermonFLLOP module, to fully exploit the potential of the Total Finite Element Tearing and Interconnecting (TFETI) domain decomposition method (DDM). Most authors researching FETI methods present weak parallel scalability with one subdomain assigned to each computational core, and call it just parallel scalability. Here we present an extension showing the data of more than one subdomain being held by each MPI process. Numerical experiments demonstrate the theoretically supported fact that for the given problem size and number of processors, the increased number of subdomains leads to better conditioning of the system operator, and hence faster convergence. Moreover, numerical, memory, strong parallel, and weak parallel scalability is reported, and optimal numbers of subdomains per core are examined. Finally, new PETSc matrix types dealing with the aforementioned extension are introduced.

A robust coarse space for optimized Schwarz methods: SORAS-GenEO-2

Optimized Schwarz methods (OSM) are very popular methods that were introduced in [11] for elliptic problems and in [3] for propagative wave phenomena. We build here a coarse space for which the convergence rate of the two-level method is guaranteed regardless of the regularity of the coefficients. We do this by introducing a symmetrized variant of the ORAS (Optimized Restricted Additive Schwarz) algorithm [17] and by identifying the problematic modes using two different generalized eigenvalue problems instead of only one as in [16,15] for the ASM (Additive Schwarz method), BDD (Balancing Domain Decomposition [12]) or FETI (Finite-Element Tearing and Interconnection [6]) methods.

On the parallel implementation of a hybrid-mixed stress formulation

This paper reports the parallel implementation of a hybrid-mixed stress finite element model for the analysis of three-dimensional structures. The stress and the displacement fields in the domain and the displacement fields on the static boundary are independently approximated. All field equations are imposed in a weighted residual form and Legendre polynomials are used to define the approximation bases. The parallel implementation of the model uses a message passing scheme based on the MPI standard. For the parallel solution of the governing system, two distinct approaches are tested. The first corresponds to the use of a third-party parallel direct solver code, named MUMPS. The second uses an in-house developed hybrid solver inspired by the FETI method and designed to deal with the typical pattern of the hybrid-mixed stress model governing system. Some numerical examples are discussed to illustrate the effectiveness and performance of the adopted parallel approach.

Two FETI-based heterogeneous time step coupling methods for Newmark and [formula omitted]-schemes derived from the energy method

The purpose of this paper is to describe a general methodology to obtain multi-time step coupling methods for different time-integration schemes (Newmark, HHT-α, WBZ-α, CH-α) on the basis of energetic considerations. The general framework of the FETI method is considered for building the heterogeneous time step coupling methods by ensuring the gluing of subdomains with Lagrange multipliers. The proposed methods for coupling subdomains with different time integration schemes and time steps (fine and large time steps), are based on the pseudo-energy measure as introduced by Hughes in the so-called energy method employed for proving the stability of implicit–explicit algorithms. Assuming a given variation of the Lagrange multipliers over the large time step, kinematic conditions are derived by ensuring the zero pseudo-energy at the interface. Two methods will be derived in this paper. The first new method, for which the interface problem involving the Lagrange multipliers is solved at the large time scale, can handle the popular dissipative α-schemes (HHT-α, WBZ-α, CH-α); in the particular case of the Newmark schemes, the new method matches with the PH method proposed by Prakash and Hjelmstad. The second new method is based at the micro time scale. It can be viewed as an extension of the method proposed by Gravouil and Combescure (GC method) for the coupling of the α-schemes. Nonetheless, like the GC method, the new micro-time-based method turns out to be dissipative when non-dissipative second-order accurate schemes are involved, and looses one order of accuracy.

Projected Krylov Methods for Solving Non-Symmetric Two-by-Two Block Linear Systems Arising from Fictitious Domain Formulations

The paper deals with the solution of large non-symmetric two-by-two block linear systems with a singular leading submatrix. Our algorithm consists of two levels. The outer level combines the Schur complement reduction with the orthogonal projectors that leads to the linear equation on subspaces. To solve this equation, we use a Krylov-type method representing the inner level of the algorithm. We propose a general technique how to get from the standard Krylov methods their projected variants generating iterations on subspaces. Then we derive the projected GMRES. The efficiency of our approach is illustrated by examples arising from the combination of the fictitious domain and FETI method.

ADVANCED DOMAIN DECOMPOSITION METHOD BY LOCAL AND MIXED LAGRANGE MULTIPLIERS

This paper presents development of an improved domain decomposition method for large scale structural problem that aims to provide high computational efficiency. In the previous researches, we developed the domain decomposition algorithm based on augmented Lagrangian formulation and proved numerical efficiency under both serial and parallel computing environment. In this paper, new computational analysis by the proposed domain decomposition method is performed. For this purpose, reduction in computational time achieved by the proposed algorithm is compared with that obtained by the dual-primal FETI method under serial computing condition. It is found that the proposed methods significantly accelerate the computational speed for a linear structural problem.

A Robin-type Domain Decomposition Method with Red-Black Partition

In this paper, we present a new Robin-type nonoverlapping domain decomposition (DD) preconditioner. The unknown variables to be solved in this preconditioned algebraic system are the Robin transmission data on the interface, which are different from the well-known DD methods like substructuring nonoverlapping DD method and FETI method. Through choosing suitable parameter on each subdomain boundary and using the tool of energy estimate, for the second-order elliptic problem, we prove that the condition number of the preconditioned system is $C(1+\log(\frac{H}{h}))^2$, where $H$ is the coarse mesh size and $h$ is the fine mesh size. Moreover, in this paper, we always assume that there is a red-black partition for the whole domain $\Omega$. Numerical results are given to illustrate the efficiency of our DD preconditioner.

High scalable non-overlapping domain decomposition method using a direct method for finite element analysis

In this paper, a novel domain decomposition method is introduced to solve a large finite element model. In this method, the decomposed domains do not overlap and are connected using a simple connective finite element, which influences the nodal point equilibrium between adjacent finite elements. This approach has the advantage that it allows use of a direct method such as Gauss elimination even in a singular problem. The singular stiffness matrices from the floating domain without the Dirichlet boundary conditions are changed into invertible stiffness matrices by assembling the connective elements. Another advantage is that computational time and storage can be reduced by using a banded matrix in the direct solver. In order to describe this proposed method, we first review the FETI method, which is the most popular domain decomposition method. Then the proposed method is introduced with a technical approach in a distributed computer system. Finally, the high scalability and computational efficiency of the proposed method are verified by comparing with the traditional FETI method for 2D and 3D finite element examples which have floating subdomains. In the result, we demonstrate high scalability for a large finite element model.

A one-level FETI method for the drift–diffusion-Poisson system with discontinuities at an interface

A 3d feti method for the drift–diffusion-Poisson system including discontinuities at a 2d interface is developed. The motivation for this work is to provide a parallel numerical algorithm for a system of PDEs that are the basic model equations for the simulation of semiconductor devices such as transistors and sensors. Moreover, discontinuities or jumps in the potential and its normal derivative at a 2d surface are included for the simulation of nanowire sensors based on a homogenized model. Using the feti method, these jump conditions can be included with the usual numerical properties and the original Farhat–Roux feti method is extended to the drift–diffusion-Poisson equations including discontinuities. We show two numerical examples. The first example verifies the correct implementation including the discontinuities on a 2d grid divided into eight subdomains. The second example is 3d and shows the application of the algorithm to the simulation of nanowire sensors with high aspect ratios. The Poisson–Boltzmann equation and the drift–diffusion-Poisson system with jump conditions are solved on a 3d grid with real-world boundary conditions.

Solving generalized eigenvalue problems on the interfaces to build a robust two-level FETI method

FETI is a very popular method, which has proved to be extremely efficient on many large-scale industrial problems. One drawback is that it performs best when the decomposition of the global problem is closely related to the parameters in equations. This is somewhat confirmed by the fact that the theoretical analysis goes through only if some assumptions on the coefficients are satisfied. We propose here to build a coarse space for which the convergence rate of the two-level method is guaranteed regardless of any additional assumptions. We do this by identifying the problematic modes using generalized eigenvalue problems.

FETI Methods for the Simulation of Biological Tissues.

In this paper we describe the application of finite element tearing and interconnecting methods for the simulation of biological tissues, as a particular application we consider the myocardium. As most other tissues, this material is characterized by anisotropic and nonlinear behavior.

국부 및 혼합 Lagrange 승수법을 이용한 영역분할 기반 유한요소 구조해석 기법 개발

In this paper, a finite element domain decomposition method using local and mixed Lagrange multipliers for a large scal structural analysis is presented. The proposed algorithms use local and mixed Lagrange multipliers to improve computational efficiency. In the original FETI method, classical Lagrange multiplier technique was used. In the dual-primal FETI method, the interface nodes are used at the corner nodes of each sub-domain. On the other hand, the proposed FETI-local analysis adopts localized Lagrange multipliers and the proposed FETI-mixed analysis uses both global and local Lagrange multipliers. The numerical analysis results by the proposed algorithms are compared with those obtained by dual-primal FETI method.

Overlapping and non-overlapping domain decomposition methods for large-scale meshless EFG simulations

Meshless methods have a number of virtues in problems concerning crack growth and propagation, strain localization, dynamic shear band propagation, projectile penetration, among others. The main drawback of such methods is that the resulting matrices are more densely populated and the computational cost for the formulation and solution of the problem is much higher than the conventional FEM. This is the reason for their limited application to academic and small-scale problems until now. In this paper, we introduce a novel approach for reducing the computational cost of meshless element free Galerkin (EFG) methods by employing domain decomposition techniques on the physical as well as on the algebraic domains. Specifically, we propose the implementation of the dual domain decomposition FETI family of methods that have been successfully used with FEM in many problems in computational mechanics. We address the issues of dividing an overlapping meshfree EFG domain to several overlapping subdomains, following the algebraic decomposition of the FETI method and alternatively on non-overlapping subdomains by modifying the required displacement compatibility conditions and by improving the accuracy of the solution through a stress update procedure.

A dual‐primal FETI method for solving a class of fluid–structure interaction problems in the frequency domain

SUMMARYThe dual‐primal finite element tearing and interconnecting method (FETI‐DP) is extended to systems of linear equations arising from a finite element discretization for a class of fluid–structure interaction problems in the frequency domain. A preconditioned generalized minimal residual method is used to solve the linear equations for the Lagrange multipliers introduced on the subdomain boundaries to enforce continuity of the solution. The coupling between the fluid and the structure on the fluid–structure interface requires an appropriate choice of coarse level degrees of freedom in the FETI‐DP algorithm to achieve fast convergence. Several choices are proposed and tested by numerical experiments on three‐dimensional fluid–structure interaction problems in the mid‐frequency regime that demonstrate the greatly improved performance of the proposed algorithm over the standard FETI‐DP method. Copyright © 2011 John Wiley & Sons, Ltd.

A theoretically supported scalable TFETI algorithm for the solution of multibody 3D contact problems with friction

A Total FETI (TFETI) based domain decomposition algorithm with preconditioning by a natural coarse grid defined by the rigid body motions of subdomains is adapted to the solution of multibody contact problems with friction in 3D and proved to be scalable for Tresca friction. The analysis admits “floating” bodies. The algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition and the discretization parameters is bounded. The analysis is based on the classical results by Farhat et al. on scalability of FETI for linear problems and on our development of optimal quadratic programming algorithms. The algorithm preserves parallel scalability of the classical FETI method. Both theoretical results and numerical experiments indicate that our algorithm is effective. The effectiveness of the method in the inner loop of the fixed point iterations for the solution of the contact problems with Coulomb’s friction is illustrated on the analysis of a mine support.

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.

A scalable TFETI algorithm for two-dimensional multibody contact problems with friction

A Total FETI (TFETI) based domain decomposition algorithm with preconditioning by a natural coarse grid of rigid body motions is adapted to the solution of two-dimensional multibody contact problems of elasticity with the Coulomb friction and proved to be scalable for the Tresca friction. The algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition parameter and the discretization parameter is bounded. The analysis is based on the classical results by Farhat, Mandel, and Roux on scalability of FETI with a natural coarse grid for linear problems and on our development of optimal quadratic programming algorithms for bound and equality constrained problems. The algorithm preserves parallel scalability of the classical FETI method. Both theoretical results and numerical experiments indicate a high efficiency of our algorithm. In addition, its performance is illustrated on analysis of the yielding clamp connection with the Coulomb friction.