Topology Optimization of Sheets in Contact by a Subgradient Method

Joakim Petersson
Department of Mechanical Engineering
Division of Mechanics
Linköping Institute of Technology
S-581 83 Linköping
Sweden

and

Michael Patriksson
Department of Mathematics
Box 354350
University of Washington
Seattle, WA 98195-4350
USA

ABSTRACT:

We consider the solution of finite element discretized optimum sheet problems 
by an iterative algorithm.  The problem is that of maximizing the stiffness 
of a sheet subject to constraints on the admissible designs and unilateral 
contact conditions on the displacements.  The model allows for zero design 
volumes, and thus constitutes a true topology optimization problem.

We propose and evaluate a subgradient optimization algorithm for a 
reformulationinto a nondifferentiable, convex minimization problem in the 
displacement variables. The convergence of this method is proven, and its low 
computational complexity is established.  An optimal design is derived through 
a simple averaging scheme which combines the solutions to the linear design 
problems solved within the subgradient method.

To illustrate the efficiency of the algorithm and investigate the properties
of the optimal designs, the algorithm is numerically tested on some
medium and large scale problems.

KEY WORDS: Optimum Sheet, Unilateral Contact, Nondifferentiable
Convex Programming, Subgradient Optimization Algorithm