The two-level FETI method for static and dynamic plate problems Part I: An optimal iterative solver for biharmonic systems

doi:10.1016/S0045-7825(97)00146-1

Computer Methods in Applied Mechanics and Engineering

Volume 155, Issues 1–2, 16 March 1998, Pages 129-151

https://doi.org/10.1016/S0045-7825(97)00146-1 Get rights and content

Abstract

We present a Lagrange multiplier based substructuring method for solving iteratively large-scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems. The proposed method is essentially an extension of the FETI domain decomposition algorithm to fourth-order problems. The main idea is to enforce exactly the continuity of the transverse displacement field at the substructure corners throughout the preconditioned conjugate projected gradient iterations. This results in a two-level FETI substructuring method where the condition number of the preconditioned interface problem does not grow with the number of substructures, and grows at most polylogarithmically with the number of elements per substructure. These theoretically proven optimal convergence properties of the new FETI method are numerically demonstrated for several finite element plate bending static and transient problems. The two-level iterative solver presented in this paper is applicable to a large family of biharmonic time-independent as well as time-dependent systems. It is also extendible to shell problems.

References (34)

C. Farhat
A Lagrange multiplier based divide and conquer finite element algorithm
J. Comput. Syst. Engrg.
(1991)
C. Farhat et al.
On a component mode synthesis method and its application to incompatible substructures
Comput. Struct.
(1994)
C. Farhat et al.
Extending substructure based iterative solvers to multiple load and repeated analyses
Comput. Methods Appl. Mech. Engrg.
(1994)
R.W. Clough et al.
Finite element stiffness matrices for analysis of plate bending
A. Gupta et al.
Optimally scalable parallel sparse Cholesky factorization
G.H. Golub et al.
Matrix Computations
(1983)
ANSYS PowerSolver
USACM-Net Digest
(Jul. 27, 1995)
R.D. Cook
Concepts and Applications of Finite Element Analysis
(1974)
D.E. Keyes et al.
J.F. Bourgat et al.
Variational formulation and algorithm for trace operator in domain decomposition calculations

P. Morice

Transonic computations by a multidomain technique with potential and Euler solvers

C. Farhat

A saddle-point principle domain decomposition method for the solution of f solid mechanics problems

C. Farhat et al.

A method of finite element tearing and interconnecting and its parallel solution algorithm

Int. J. Numer. Methods Engrg.

(1991)

C. Farhat et al.

An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems

SIAM J. Sci. Stat. Comput.

(1992)

J. Mandel

Balancing domain decomposition

Comm. Appl. Numer. Methods

(1993)

P. LeTallec et al.

Balancing domain decomposition for plates

C. Farhat et al.

Optimal convergence properties of the FETI domain decomposition method

Comput. Methods Appl. Mech. Engrg.

(1994)

Cited by (136)

A domain decomposition strategy for mCRE-based model updating in dynamics
2023, Computer Methods in Applied Mechanics and Engineering
In structural dynamics, the modified constitutive relation error is a widely used functional for model updating. The method involves a two-step iterative process to minimize the functional: first, regions with high modeling errors are localized, then their associated parameters are updated. The localization step requires solving large linear systems at each iteration, which makes the method less suitable for industrial models. In this work, a domain decomposition formulation is proposed. It makes the method more flexible, as subdomains can use different models which are independently modified during the updating algorithm without any additional cost. Moreover, this approach is very efficient numerically as it benefits from the same advantages as domain decomposition methods. In its standard form, it lacks a coarse problem which seems problematic; this is however addressed by using a multi-preconditioned solver such as MPGMRES. The performance of the proposed approach is illustrated by means of academic and industrial examples, with practical implementation in a commercial software.
Asynchronous multisplitting-based primal Schur method
2023, Journal of Computational and Applied Mathematics
This paper presents a new scalable asynchronous domain decomposition method which significantly outperforms the recent two-level asynchronous restricted additive Schwarz (RAS) method. The novel approach consists of considering the classical primal Schur complement framework and solving the interface problem using a multisplitting relaxation based on a weighted additive Schwarz preconditioner. The computational model does not require any significant implementation effort compared to an asynchronous RAS solver. Experiments were conducted on a linear elasticity test case, and the results clearly show the practical superiority of the new asynchronous method, which is even competitive with its conjugate gradient counterpart.
Accelerating the convergence of AFETI partitioned analysis of heterogeneous structural dynamical systems
2020, Computer Methods in Applied Mechanics and Engineering
Variationally based algorithms for the partitioned solution of structural mechanics problems are presented. Two key features of the present algorithms are the judicious application of the d’Alembert-Lagrange principal equations and the use of dominant substructural deformation modes. The paper includes three developments:
1. Variational derivation of AFETI parallel solution methods.
2. One-level and two-level AFETI implicit transient analysis algorithms with coarse problem included in the projector and based on free floating rigid body modes.
3. A new AFETI implicit transient solution algorithm derived by constraining the interface equilibrium equations with the floating and dominant deformational modes.
In addition to variational derivations of solution algorithms, the present paper is strived to offer new physical and/or numerical insight as each of variational derivational steps is succinctly explained. Performance evaluations of the algorithms described herein are presented.
An iterative multiscale modelling approach for nonlinear analysis of 3D composites
2018, International Journal of Solids and Structures
The advent of new more complex classes of strongly heterogeneous materials, such as 3D woven composites, introduces new challenges for well-established finite element multiscale modelling approaches. These materials' internal architecture dominates the local stress concentrations, damage initiation and damage progression. Additionally, the material loses periodicity during manufacture and conventional homogenization approaches become inapplicable. In this paper, a multiscale modelling approach based on domain decomposition and homogenization is proposed to model the mechanical behaviour of these materials. The proposed model formulates a set of displacement and force compatibility conditions between the various subdomains. The compatibility conditions are formulated by limiting the set of kinematically admissible solutions on the smaller scales in order to satisfy the larger scale basis functions at the interfaces. The different subdomains are solved alongside the compatibility conditions in an iterative process. The proposed multiscale framework reduces the stress artefacts on the subdomains' boundaries. This feature allows the 3D woven material internal architecture represented at the smaller scales to control the structural response at the global scale. Additionally, this framework allows for selective application of nonlinear material models to the subdomains of interest through its ability to redistribute stresses across the subdomains boundaries.
Parallel computation for three-dimensional shell analysis of curved configuration based on domain decomposition method
2018, Journal of Computational Science
In this paper, a parallel computational algorithm is developed based on finite element tearing and interconnecting (FETI) method, specifically, localized Lagrange multipliers. The proposed FETI method decomposes large-size structures into non-overlapping subdomains via localized Lagrange multipliers. To consider the curved configuration of large-size structures, a facet shell element created by combining an optimal triangle membrane and discrete Kirchhoff triangle bending plate (OPT-DKT) is suggested and used by introducing rotational operators. Moreover, practical performance of the present OPT-DKT facet shell element is evaluated through static and dynamic analysis. Finally, parallel computation is implemented for the proposed approach using message passing interface (MPI).
Numerical analysis of coupled heat and moisture transport in masonry
2017, Computers and Mathematics with Applications
Citation Excerpt :
The original method is denoted FETI-1. In 1998, so called FETI-2 was introduced in [7,8] and FETI-DP was published in [9]. FETI method with all floating subdomains was described in [10].
Coupled analysis of heat and moisture transport in real world masonry structures deserves a special attention because the spatial discretization by the finite element method leads usually to large number of degrees of freedom. Thin mortar layers and large bricks or stones have very different material properties and the finite element mesh has to be able to describe correct temperature and moisture fields in mortar and in its vicinity in the blocks. This paper describes two possible solutions of such problems. The first solution is based on the domain decomposition method executed on parallel computers, where the Schur complement method is used with respect to non-symmetric systems of algebraic equations. The second alternative method is the application of a multi-scale approach in connection with a processor farming method, where the whole structure is described by a reasonably coarse finite element mesh, called the macro-scale problem, and the material parameters are obtained from the lower-level problems, called the meso-scale problem, by a homogenization procedure. In this procedure, the macro-problem is assigned to the master processor and the meso-scale problems belong to the slave processors in the processor farm.

View all citing articles on Scopus

View full text

The two-level FETI method for static and dynamic plate problems Part I: An optimal iterative solver for biharmonic systems

Abstract

J. Comput. Syst. Engrg.

Comput. Struct.

Comput. Methods Appl. Mech. Engrg.

Optimally scalable parallel sparse Cholesky factorization

Matrix Computations

USACM-Net Digest

Concepts and Applications of Finite Element Analysis

Variational formulation and algorithm for trace operator in domain decomposition calculations

Transonic computations by a multidomain technique with potential and Euler solvers

A saddle-point principle domain decomposition method for the solution of f solid mechanics problems

A method of finite element tearing and interconnecting and its parallel solution algorithm

Int. J. Numer. Methods Engrg.

An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems

SIAM J. Sci. Stat. Comput.

Balancing domain decomposition

Comm. Appl. Numer. Methods

Balancing domain decomposition for plates

Optimal convergence properties of the FETI domain decomposition method

Comput. Methods Appl. Mech. Engrg.