## 74B20 Nonlinear elasticity

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A cubic regularization algorithm for nonconvex optimization in function space (in preparation)
(2012)

We propose a cubic regularization algorithm that is constructed to deal with nonconvex minimization
problems in function space. It allows for a flexible choice of the regularization term and thus accounts for the fact
that in such problems one often has to deal with more than one norm. Global and local convergence results are established
in a general framework. Moreover, several variants of step computations are compared. In the context of nonlinear
elasticity it turns out the a cg method applied to an augmented Hessian is more robust than truncated cg.

We consider a shape implant design problem that arises in the context of facial surgery.
We introduce a reformulation as an optimal control problem, where the control acts
as a boundary force. The state is modelled as a minimizer of a polyconvex
hyperelastic energy functional. We show existence of optimal solutions and
derive - on a formal level - first order optimality conditions. Finally, preliminary numerical results
are presented.

The paper extends affine conjugate Newton methods from convex to nonconvex minimization, with particular emphasis on PDE problems originating from compressible hyperelasticity. Based on well-known schemes from finite dimensional nonlinear optimization, three different algorithmic variants are worked out in a function space setting, which permits an adaptive multilevel finite element implementation. These algorithms are tested on two well-known 3D test problems and a real-life example from surgical operation planning.