## A Feasible Sequential Convex Programming Method for Free Material
Optimization

S. Ertel, K. Schittkowski, C. Zillober: 8th World Congress on Structural and
Multipdisciplinary Optimization, June 1-5, 2009, Lisbon, Portugal (2009)

**Abstract:**
In free material optimization (FMO), one tries to find the best mechanical
structure by minimizing the weight or by maximizing the stiffness with
respect to given load cases. Design variables are elasticity tensors or
elementary material matrices, respectively, based on a given FE discretization.
Material properties are as general as possible, i.e., anisotropic, leading to
positive definite elasticity tensors, which may be arbitrarily small in case of
vanishing material. To guarantee a positive definite global stiffness matrix for
computing design constraints, e.g., displacements or stress constraints, it is
required that all iterates of an optimization algorithm retain strictly positive
definite tensors. We propose a modification of a sequential convex programming method
(SCP). The corresponding separable and strictly convex nonlinear subproblems are
expanded by additional nonlinear constraints, to guarantee positive definite
elementary matrices in each iteration step. These constraints are retrieved from
a sign condition of sub-determinants. The algorithm is introduced and some
numerical examples based on academic test cases are presented.