An Optimal Control Problem in Polyconvex Hyperelasticity

Abstract

Abstract. We consider an implant shape design problem arising in the context of facial surgery. The aim is to find the shape of an implant that deforms the soft tissue of the skin in a desired way. Assuming sufficient regularity, we introduce a reformulation as an optimal control problem where the control acts as a boundary force. The solution of that problem can be used to recover the implant shape from the optimal state. For a simplified problem, in the case where the state can be modeled 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 for the original optimal control formulation.

Optimal control of static contact in finite strain elasticity

“…Additionally, the total energy functional I is non-convex and therefore, its minimizers do not have to be unique. In [20], the existence of solutions to an optimal control problem in hyperelasticity without contact constraints has been proven. We can directly transfer the results from [20] to our analysis.…”

“…In many cases, not only simulation of elastic bodies is of interest, but also optimization problems in the context of elasticity may be considered. In [18,20], the authors studied the design of implants which can be modeled by an optimal control problem of a hyperelastic body using a tracking type objective functional. In these works, a rigorous proof for the existence of optimal solutions to such kinds of problems was elaborated for the first time.…”

An affine covariant composite step method for optimization with PDEs as equality constraints

“…Assume that x k converges to the SSC point x * in the setting described above. Assume further that the Lipschitz conditions (27), (28), and (29) hold in a neighborhood of x * . Then its convergence is superlinear.…”

“…Now we consider a simplified example from implant shape design with control and observation on disjoint parts of the boundary [27]. Thus we work with the same function spaces for Y and P and replace the control space by U = L 2 (Γ c ), where Γ c denotes the control boundary.…”

Weakly symmetric stress equilibration for hyperelastic material models

“…Expressing the internal forces of a material, the components of the stress‐tensor are crucial for the prediction of the weakening of a material, including plastic behavior or damage. A specific application area where this is an issue is associated with implant shape design which constitutes an optimal control problem . Therefore, the accurate approximation of the stress‐tensor is of strong importance in numerous applications and in particular in the hyperelastic material model this paper is concerned with.…”