ISBN 9783868534740

978-3-86853-474-0, Reihe Mathematik

Wolfgang Hess
Geometry Optimization with PDE Constraints and Applications to the Design of Branched Sheet Metal Products

163 Seiten, Dissertation Technische Universität Darmstadt (2010), Softcover, B5

Zusammenfassung / Abstract

This work is concerned with algorithms for the solution of partial differential equations (PDE) constrained optimization problems which arise in the shape optimization of steel profiles. Today, a wide variety of sheet metal products is used in many different industries. Roll forming is an important manufacturing process for the volume production of these products. In combination with other processes such as laser beam welding and linear flow splitting, steel profiles can be manufactured from a single piece of sheet metal. As part of an algorithmic product development, we want to optimize the geometry of sheet metal products, given a topology and starting geometry. We present an optimization model that uses the three-dimensional linear elasticity equations as a PDE constraint to model the mechanical behavior of the product. The mean square norm of the displace- ment is used as the objective function. Our approach to tackle the arising optimization problems can be applied directly to problems with different PDE constraints.

Under suitable assumptions, we show differentiability of the weak formulation of the PDE with respect to the design and use adjoint solutions to compute gradients of the reduced objective function. We give the symbolic derivatives of the finite element approximation of the PDE and also examine the use of automatic differentiation. To- gether with quasi-Newton updates for the objective Hessian, we obtain the necessary derivative information to use sequential quadratic programming (SQP) methods.

We prove convergence of an inexact SQP method which we use to control the inex- actness of the iterative solution of the state and adjoint equations and show the suitabil- ity of the symmetric rank-1 (SR1) quasi-Newton update with inexact objective gradient evaluations. In order to compute a solution of the PDE constrained optimization prob- lem, we use nested iterations of the inexact SQP method. For each outer iteration, a so- lution to the approximate optimization problem for a fixed reference mesh is computed until a stopping criterion is satisfied. Then the mesh is refined taking into account both the state and the adjoint error. We examine the convergence of the nested iterations for three different stopping criteria of the inner iterations.

Rezensionen

Zentralblatt MATH, 1209.90003, pdf