# Methods for Large-Scale Variable Thickness Sheet Problems

Mathias Stolpe^{*}

The optimal structural design problem which we consider is the well-studied variable thickness sheet (VTS) problem. In this problem the structural stiffness is maximized with a constraint limiting the volume of the structure. The presentation shows that by using modern numerical methods it is possible to solve large-scale VTS problems over 3D design domains using a modest number of function evaluations. Although the VTS problem has only marginal industrial interest it shares many of the challenges of other structural topology optimization problems which are frequently solved. The proposed techniques may thus prove useful when developing numerical methods for other structural topology optimization problems. Our choice of optimization method is a primal-dual interior point method recently proposed for general nonlinear optimization problems. Our implementation does not require that the stiffness matrix or the primal-dual saddle-point systems are assembled or factorized. Instead, these are only involved in matrix-vector multiplications as part of Krylov subspace methods for performing the structural analysis and for computing the search direction in the interior point method. The main difficulty in applying Krylov subspace methods for solving the saddle-point system in interior point methods is the inherent and ill-conditioning of the saddle-point matrix as the optimum is approached. The structural stiffness matrix also experience increasing ill-conditioning as the optimization method modifies the design. We propose preconditioners for both systems and numerically show that they are powerful enough to deal with the increasing ill-conditioning. An extensive set of numerical experiments on 3D design domains suggest that the combination of techniques result in a robust and efficient method capable of solving large-scale instances of the VTS problem.

Mathematics Subject Classification: 74P05 90C90 90C51

Keywords: Structural topology optimization; interior point method; Krylov subspace method; preconditioners

Minisymposion: Material and Topology Optimization: Theory, Methods and Applications