The most challenging problem when performing implicit analysis is to solve large systems of linear algebraic equations in parallel. When a refined Mindlin-Reissner theory is employed for description of each layer of a laminated composite material, the matrix of resulting system of equations is sparse and exhibits regular sparsity pattern. However, it is easy to observe that its solution by a variant of the Gauss elimination is not efficient due to the fill-in . For example, only the 6-layer composite plate discretized by mesh of 16x16 elements may be treated on a computer with 128 MB memory. Moreover, since the matrix of the system is indefinite, it is difficult to find an efficient preconditioner for application of the standard iterative methods.
A natural discretization of the equations of equilibrium and kinematic constraints is shown to result in a system of linear equations whose structure resembles that of the Finite Element Tearing and Interconnecting (FETI) method for some other problems. A variant of the FETI method is then proposed to the solution of the system.
The FETI method was introduced by Farhat and Roux in . The idea of the method is based on elimination of displacements from equilibrium conditions and substituting them to the compatibility conditions which are expressed with the help of localization matrices. Non-supported subdomains require the definition of pseudoinverse matrices. The rigid body motions of the non-constrained subdomains are used for global data interchange. The linear combination of rigid body motions must be added to the displacements with respect to solvability conditions on particular subdomains.
The FETI method was implemented in parallel environment and successfully applied for solution of composite laminated plates and shells. To demonstrate its capability, the analysis of a composite laminated plate with layup is presented. The unstructured mesh of the domain (single layer) is illustrated in Fig. 19 The domain is clamped on the external perimeter and subjected to point load at the ``corner'' nodes. A special layer-based domain decomposition is adopted resulting in a separate domain for each layer.
Each of the 6 plies of the composite is made of aligned T-50 graphite fibers bounded to the 6061 Aluminum matrix with volume fraction 0.5. The individual material properties are listed in Table 3. The overall properties are obtained by the Mori-Tanaka averaging method [10,11].
The calculation was performed on PC cluster using 6 processors. The dependence of the number of iterations on number of DOFs is sketched in Fig. 20. Note, that with increasing size of the problem the number of iterations grows only polylogarithmically. The deformed shape of the plate is depicted in Fig. 21.