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Parallel Implicit Analysis of Composite Plates

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