Applying Deflation Methods in a Topology Optimization Procedure

Abstract

Structure Optimization has been an important subject with many applications for centuries. In the last sixty years, numerical optimization has facilitated large advancements in this field. One of the areas in Structure Optimization is Topology Optimization, which is used for Additive Manufacturing purposes. In this thesis we explore Static and Dynamic Topology Optimization. In the optimization problems matrix equations of the type Ax = b, with A sparse and badly conditioned, are accelerated using deflation techniques in addition to preconditioning. We have applied several iterative methods, preconditioners, and deflation types to the topology optimization problems. The static problem concerns compliance optimization of a two-dimensional MBB-beam. The deflation type that reduced the number of iterations the most without introducing large overhead costs was rigid body modes deflation, divided over element squares. It was found that dividing the rigid body modes vectors over density based regions did not reduce the number of iterations. The dynamic problem concerns eigenvalue optimization of a three-dimensional moving wafer stage that is used for laser-printing computer chips. The optimization formulation contains a shifted eigenvalue problem that is solved using model order reduction. In the computations of bases for the reduction matrix equations appear, to which deflation techniques were applied. All deflation types reduced the number of iterations and needed time to solve the matrix equations. The best deflation type was using rigid body modes (RBM) divided over element cubes combined with eigenvectors from the previous iteration. The next best was the same combination without the division over cubes. Using eigenvectors or RBM separately were the least effective deflation types. There is an optimal amount of element cubes to use when dividing the RBM. The tests on a few grid sizes suggested a quadrupling of the amount when the grid size doubles, but more research is needed to really identify a relation between the grid size and optimal amount. When increasing the grid size to a level where parallel computing on a cluster was required, the deflation type using element cubes could not be used due to complications with parallel implementation. Using the deflation type RBM and eigenvectors, reductions by a factor of 1.60 and 1.75 in the total needed time for 150 optimization iterations were achieved for grid sizes 120x120x20 and 180x180x30, respectively. In the first case the objective function of the optimized design converged further in the same amount of iterations when using deflation. future research could include using the promising deflation type RBM in cubes + eigenvectors in parallel computing to obtain an even larger time reduction in the optimization of the wafer stage with large grid sizes.