In this talk we discuss an optimal control problem with a quadratic cost functional, which is governed by a semi-linear elliptic equation with a non-monotone nonlinearity. Although the state equation as well as its linearization are in general not wellposed, we provide existence results and derive optimality conditions for the optimal control problem. The theory includes localized controls and provides an optimality system in the qualified form. The existence of an adjoint state is given in a constructive way suitable for an algorithmic approach.

A common procedure in isogeometric shape optimization is to use a NURBS description of the geometry and consider a subset of control point coordinates as design variables [1, 2, 3], which corresponds to the first discretize–then optimize approach. While this is a fairly straightforward way to perform shape optimization in the context of isogeometric analysis, the results will usually depend on the particular representation of the geometry, since the problem is reduced to a finite-dimensional one that often disregards the inherent geometric structure of shapes. We present the Riemannian framework for PDE constrained shape optimization [4] in an attempt to provide suitable descent paths for control points that evolve the shape in a consistent manner independent of the chosen representation. The idea is to endow the shape space with a reparametrization-invariant Riemannian metric [5] and move the shapes along geodesic paths corresponding to the metric, which allows for mesh-independent shape optimization. This first optimize–then discretize approach has been applied to the compliance minimization of thin elastic shells using a first-order Sobolev-type metric in [6]. We now investigate the effects of using geodesic retractions for optimization on Riemannian shape spaces of planar curves and surfaces, where isogeometric finite element methods are used for the discretization of the equations to efficiently obtain accurate numerical approximations of the Riemannian shape gradient.

References:

[1] W. Wall, M. Frenzel, and C. Cyron. Isogeometric structural shape optimization. Computer Methods in Applied Mechanics and Engineering, 197:2976–2988, 2008.

[2] K.-U. Bletzinger, J. Kiendl, R. Schmidt, and R. Wüchner. Isogeometric shape optimization of shells using semi-analytical sensitivity analysis and sensitivity weighting. Computer Methods in Applied Mechanics and Engineering, 274:148–167, 2014.

[3] J. Lopez, C. Anitescu, and T. Rabczuk. Isogeometric structural shape optimization using automatic sensitivity analysis. Applied Mathematical Modelling, 89:1004–1024, 2021.

[4] V. H. Schulz, M. Siebenborn, and K. Welker. PDE constrained shape optimization as optimization on shape manifolds. Geometric Science of Information, volume 9389 of Lecture Notes in Computer Science, pages 499–508. Springer, 2015.

[5] M. Bauer, P. Harms, and P. W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 3(4):389–438, 2011.

[6] R. Rosandi and B. Simeon. Riemannian shape optimization of thin shells using isogeometric analysis. Proceedings in Applied Mathematics and Mechanics, 25(1):e202400204, 2025.

Light elastic structures arise naturally in the optimization of designs that carry prescribed loads with vanishing material volume. In this talk, I discuss a relaxed shape-optimization problem for such structures, formulated in terms of a stress measure and a material distribution subject to an equilibrium constraint. The model is motivated by the theory of Michell trusses and by recent Γ-convergence results for elastic energies in the vanishing-mass regime.

I will present existence of minimizers for the relaxed problem and derive first-order optimality conditions using a Lagrangian formulation. The resulting Karush-Kuhn-Tucker conditions provide a measure-theoretic description of the alignment principle for optimal trusses, as studied by Bouchitté and Buttazzo via the Monge–Kantorovich equation and further developed by Bouchitté, Gangbo, and Seppecher in the context of Michell trusses and lines of principal action. I will then show that these conditions provide geometric information on optimal designs. In two space dimensions, the singular part of the optimal stress is concentrated on a countably one-rectifiable set, while in three dimensions both one- and two-dimensional rectifiable components may occur.

This is joint work with Dimitrios Andreakis and Filip Rindler.