Keywords: FETI method, domain decomposition, parallel computing, contact problems, composite materials.

This paper describes the family of FETI methods and their applications in various engineering and scientific problems. In 1991, Farhat and Roux introduced the finite element tearing and interconnecting method, abbreviated to FETI, which became one of the most popular non-overlapping domain decomposition methods. In time, several derivatives of the original method were introduced for particular problems. The original method is denoted the one-level FETI method. The two-level FETI method was introduced for the solution of plate and shell problems. Dostál and coworkers introduced the total FETI method, abbreviated as TFETI. For the solution of the Helmholtz problems, the FETI-H method was introduced.

The one-level FETI method is based on decomposition of the original domain into smaller subdomains and the continuity condition on the subdomain interfaces is enforced by Lagrange multipliers. The original unknowns are eliminated and the coarse problem, containing unknown Lagrange multipliers and coefficients of combinations of the rigid body modes, is obtained. The decomposition of the domain can lead to floating subdomains which are not constrained and it results in singular subdomain matrices. Therefore, the pseudo-inverse matrices instead of the inverse ones have to be used. The coarse problem is solved using the modified conjugate gradient method because its matrix is generally positive semidefinite.

The TFETI method decomposes the original domain into subdomains but simultaneously it releases all constraints. Their effect is enforced by Lagrange multipliers. The main advantage of the method is based on the known number of the rigid body modes in advance. Moreover, the rigid body modes can be determined explicitly. The subdomains are virtually fixed in selected fixing nodes which cause nonsingular subdomain matrices with a reasonable condition number. The matrices are factorised and remaining rows and columns which correspond to the unknowns defined in the fixing nodes are processed by the singular value decomposition algorithm. This approach is much more robust than those previously used.

In this paper, application of the TFETI method to contact problems based on the optimal algorithm SMALBE-M is described. The multipliers are defined on the subdomain interfaces, a part of the boundary with the Dirichlet boundary condition and on the contact zone. The FETI method was successfully applied to analysis of composite materials. The perfect and imperfect bond between fibres and composite matrix is modelled. Selected real world problems demonstrate the possibilities of FETI methods.

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