Appl Math Optim 44:227–244 (2001)
2001 Springer-Verlag New York Inc.
On the Stability of Sizing Optimization Problems
for a Class of Nonlinearly Elastic Materials
Institut f¨ur Industriemathematik, Universit¨at Linz,
A-4040 Linz, Austria
Abstract. In this work we deal with a stability aspect of sizing optimization prob-
lems for a class of nonlinearly elastic materials, where the underlying state problem
is nonlinear in both the displacements and the stresses. In  it is shown under
which conditions there exists a unique solution of discrete design problems for a
body made of the considered nonlinear material, if the nonlinear state problem is
solved exactly. In numerical examples the nonlinear state problem has to be solved
iteratively, and therefore it can be solved only up to some small error ε.
The question of interest is how this affects the optimal solution, respectively
the set of solutions, of the design problem. We show with the theory of point-to-set
mappings that if the material is not too nonlinear, then the optimal design depends
continuously on the error ε.
Key Words. Structural optimization problem, Nonlinear state problem, Stability,
AMS Classiﬁcation. 49B50, 93C20.
The nonlinear elastic material law we consider in this paper is essentially Hooke’s law
(see pp. 162ff of ), but Young’s modulus E depends on the stress ﬁeld σ, i.e.,
E = E (σ); (1)
e.g., the gray cast iron GGL25 that is used in car engines fulﬁlls (1) (see also Section 6).
This work was partially supported by the Christian Doppler Society (Austria).