Keywords: FETI method, domain decomposition, parallel computing.

This contribution deals with the FETI method and its applications. In 1991, Farhat and Roux published the paper [1] about Finite Element Tearing and Interconnecting method. It is a non-overlapping domain decomposition method suitable for parallel computing. The original domain is split into several smaller subdomains. The internal unknowns, shared by only one subdomain, are eliminated while the interface unknowns, shared by at least two subdomains, contribute to the coarse problem. The continuity on interfaces is enforced by Lagrange multipliers. The coarse problem is solved by a modified conjugate gradient method because the matrix of the system is positive semidefinite. Special attention has to be devoted to subdomain matrices which are generally singular. The rigid body modes of the substructures are needed for construction of the coarse problem.

Several versions of the original method were published. The dynamic problems require a modification because the dynamic stiffness matrix is always non-singular and the classical coarse problem cannot be assembled. Another modification was motivated by a poor convergence of the method for the fourth-order problems. Additional Lagrange multipliers are defined at the corners and an augmented system of equations is obtained.

The FETI method is a very suitable tool for the solution of contact problems. Especially the knowledge of rigid body modes significantly helps the efficient solution in comparison with other approaches. The method was used for frictionless contact problems as well as problems with the Coulomb friction. Application of the FETI method to composite materials has been very successful. The method efficiently solves delamination problems in fibre composites. The application of the FETI method to analysis of composite laminated plates leads to a significant speedup.

Two basic preconditioners were introduced, the mathematically optimal Dirichlet preconditioner and the economical lumped preconditioner. It was observed numerically and later proved mathematically that the condition number of the matrix of the coarse problem in the preconditioned modified conjugate gradient method depends on the ratio between the size of subdomain and the size of finite element. The condition number grows only polylogarithmically which results in numerical scalability of the method. The parallel scalability was studied and proved on a cluster with 1000 processors.

The FETI method together with the substructuring method were blended and a new domain decomposition method, called FETI-DP, was introduced in 2000. The FETI-DP method is the state-of-the-art method in domain decompositions at this time.

[1]

C. Farhat, F.X. Roux, "A Method of Finite Element Tearing and Interconnecting and its Parallel Solution Algorithm", International Journal for Numerical Methods in Engineering, 32, 1205-1227, 1991. doi:10.1002/nme.1620320604

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