A Recursive Trust-Region Method for Non-Convex Constrained Minimization

Abstract

The mathematical modelling of mechanical or biomechanical problems involving large deformations or biological materials often leads to highly nonlinear and constrained minimization problems. For instance, the simulation of soft-tissues, as the deformation of skin, gives rise to a highly non-linear PDE with constraints, which constitutes the first order condition for a minimizer of the corresponding non-linear energy functional. Besides of the pure deformation of the tissue, bones and muscles have a restricting effect on the deformation of the considered material, leading to additional constraints. Although PDEs are usually formulated in the context of Sobolev spaces, their numerical solution is carried out using discretizations as, e.g., finite elements. Thus, in the present work we consider the following finite dimensional constrained minimization problem:

$$u \in {\rm B}: J(u) = \min !$$

(M)

where $B = \{\nu \in {\mathbb R}^n\,\, | \,\, {\underline \varphi} \leq \nu \leq {\overline \varphi} \}$ and $\underline \varphi < {\overline \varphi} \} \,\, \in \,\, {\mathbb R}^n$ and the possibly nonconvex, but differentiable, objective function ${\bf \it J} \, : \, \mathbb R^{n} \rightarrow \mathbb R.$. Here, the occurring inequalities are to be understood pointwise. In the context of discretized PDEs, n corresponds to the dimension of the finite element space and may therefore be very large.