| Author | Muralidharan Pandheeradi |
|---|---|
| Title | Fast, Efficient Multi-Level Iterative Solution Techniques for Large-Scale Linear/Nonlinear Finite Element Systems |
| Year | 1995 |
| School | Civil Engineering |
| Institution | RPI |
| Abstract | The thesis deals with one of the challenging issues in the area of application of finite element methods to the solution of problems in Mechanics, namely the efficient solution of the large systems of equilibrium equations arising from finite element discretization. Two important classes of problems of comparable complexity are addressed, one each in the areas of linear and nonlinear case. A fast, efficient multilevel solution scheme is developed for large systems of nonlinear equations arising from finite element discretization. The proposed solver results from an innovative combination of the BFGS quasi-Newton method and FAS version of the multigrid method, that preserves the desirable features of both the methods while minimizing most of their significant drawbacks. The numerical experiments on history-dependent, elasto-plastic problems demonstrate the potential of the new solution methodology for large-scale nonlinear systems. The problem considered in the linear case belongs to the class of nonpositive definite symmetric systems resulting from hybrid formulations that weakly enforces compatibility and traction continuity conditions between independently modeled substructures. Two distinct approaches are investigated - the domain decomposition method and the global-local type method. In the domain decomposition approach, the nonpositive definite system is transformed into an equivalent positive definite system that can be solved by either iterative or direct method without pivoting. A multigrid-like algorithm is developed that uses the collocation problem as the auxiliary grid, and a new, computationally efficient preconditioner for smoothing. The superiority of the two-level iterative scheme is highlighted by numerical examples, including that of a Boeing crown panel. In the global-local approach, the hybrid system is decomposed into a hierarchical global-local problem consisting of a positive definite global problem and an indefinite local system that are derived using minimization on the subspace and the stationarity principle, respectively. Numerical performance studies for two-dimensional plane stress and shell problems verify the usefulness of the proposed procedures. |