** Next:** Conclusions
**Up:** Examples
** Previous:** Nonlinear Parallel Analysis by Explicit Algorithm

In contrast to explicit methods, implicit algorithms require solution of system of algebraic equations.
Since there occur many degrees of freedom, thanks to complexity and
size of the models, usual hardware is not able to
solve those huge systems. This makes parallel computers together with
parallel algorithms very promising.
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 [13].
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.
The refined Reissner-Mindlin theory in connection with the finite element method leads to special
form of the resulting matrix which is created from blocks. However, one diagonal block contains only zero
components and this causes difficulty for solution methods. Similar matrix occurs in Finite Element
Tearing and Interconnecting method (FETI).
The FETI method was introduced by Farhat and Roux in [9].
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.
Excellent behaviour was observed in analysis of composite laminated plates. The
most interesting result consists in the fact that almost the same number of
iterations was evidenced for rectangular plate with 120.000 degrees of
freedom and more as it is depicted in Fig. 18.
The analysis of another composite laminated plate with layup
is presented as the second example. 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 2. 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.

** Next:** Conclusions
**Up:** Examples
** Previous:** Nonlinear Parallel Analysis by Explicit Algorithm
*Daniel Rypl *

2005-12-03