Generalized Aggregation Multilevel Solver

The paper presents a Generalized Aggregation Multilevel (GAM) solver, which automatically constructs nearly optimal auxiliary coarse models based on the information available in the source grid only. GAM solver is a hybrid solution scheme where approximation space of each aggregate (group of neighboring elements) is adaptively and automatically selected depending on the spectral characteristics of individual aggregates. Adaptive features include automated construction of auxiliary aggregated model by tracing “stiff” and “soft” elements, adaptive selection of intergrid transfer operators, and adaptive smoothing. An obstacle test consisting of nine industry problems, such as ring-strut-ring structure, casting setup in airfoil, nozzle for turbines, turbine blade and diffuser casing as well as on poor conditioned shell problems, such as High Speed Civil Transport, automobile body and canoe, was designed to test the performance of GAM solver. Comparison to the state of the art direct and iterative (PCG with Incomplete Cholesky preconditioner) is carried out. Numerical experiments indicate that GAM solver possesses an optimal rate of convergence by which the CPU time grows linearly with the problem size, and at the same time, robustness is not compromised, as its performance is almost insensitive to problem conditioning. 1.0 Introduction The performance of linear solvers in terms of CPU time for symmetric positive definite systems can be approximated as , where N is the number of degrees-of-freedom, and C, β are solution method dependent parameters. The major advantage of direct solvers is their robustness, which is manifested by the fact that parameters C and β are independent of problem conditioning (except for close to singular systems). Direct solvers are ideal for solving small up to medium size problems since the constant C for direct methods is significantly smaller than for iterative solvers, but becomes prohibitively expensive for large scale problems since the value of exponent for direct solvers is higher than for iterative methods. To make direct solvers more efficient various modifications of Gaussian eliminaCN β