The optimal design of medical and dental implants, or lightweight structures in aeronautics can be modelled by a periodic structure with an empty, but a-priorily unknown inclusion. Homogenisation of this periodic scaffold structure, i.e., a material containing periodically arranged, identical copies of a cavity, leads to a macroscopic equation involving an effective material tensor $\mathbf{A}_0(\Omega) \in \mathbb{R}^{d \times d}_{\rm sym}$.

This effective tensor is determined by a microscopic problem, defined on the $d$-dimensional, periodic unit cell $Y := [- \frac{1}{2}, \frac{1}{2} ]^d$, containing the void $\Omega \subset Y$. The solutions of the respective cell problems \[ \begin{cases} \Delta w_i = 0 &\text{in } Y \setminus \overline{\Omega}, \\ \partial_{\boldsymbol{n}} w_i = - \langle \boldsymbol{n}, \, \boldsymbol{e}_i \rangle &\text{on } \partial \Omega, \end{cases} \qquad i = 1, \ldots, d, \] define the coefficients of the effective tensor by \[ a_{i, j}(\Omega) = \int_{Y \setminus \overline{\Omega}} \big\langle \boldsymbol{e}_i + \nabla w_i, \ \boldsymbol{e}_j + \nabla w_j \big\rangle \operatorname{d}\!\boldsymbol{y}. \] Therefore, effective material tensor on the macroscopic scale is given by the solution of a problem on the microscopic scale.

Considering a sought material tensor $\mathbf{B} \in \mathbb{R}^{d \times d}_{\rm sym}$, which expresses desired material properties, we may ask the following question: Can we find a cavity $\Omega$ such that the effective tensor is as close to $\mathbf{B}$ as possible? In other terms, we want to minimise the tracking type functional \[ J(\Omega) := \frac{1}{2} \big\| \mathbf{A}_0(\Omega) - \mathbf{B} \big\|_F^2. \]

In [2], formulae for the shape gradient of the functional $J(\Omega)$ have been derived and numerical examples in two dimensions were presented, whereas in [3], integral equations were used to obtain numerical results in three dimensions. These examples include simply connected cavities and also more complex cavities of genus greater than zero. The calculations were performed with the isogeometric C++ library BEMBEL [1].

[1] J. Dölz, H. Harbrecht, S. Kurz, M. Multerer, S. Schöps, F. Wolf. Bembel: The fast isogeometric boundary element C++ library for Laplace, Helmholtz, and electric wave equation, SoftwareX, 11: 100476, 2020.

[2] M. Dambrine, H. Harbrecht. Shape optimization for composite materials and scaffold structures, Multiscale Modeling & Simulation, 18: 1136--1152, 2020.

[3] H. Harbrecht, M. Multerer, R. von Rickenbach. Isogeometric shape optimization of periodic structures in three dimensions, Computer Methods in Applied Mechanics and Engineering, 391: 114552, 2022.

Numerically solving shape optimization problems usually takes advantage of the Zolesio-Hadamard form, which writes the shape derivative of the objective function into a boundary integral of the product of the shape grdient and the deformation field. Intuitively, one can choose the deformation field to take the form of the negative of the shape gradient, evaluated on the free boundary, as a gradient descent direction. However, such choice may cause instabilities and oscillations on the free boundary. This issue is a motivation for extending the deformation field to the computational domain in a smooth manner, this method is known as the traction method [1]. In this talk, solenoidal extensions to solve shape optimization problems with volume constraints will be considered. In particular, the deformation field will be extended to the computational domain by solving an incompressible Stokes equations with a Robin data defined as the negative of the shape gradient and the viscosity constant assumed to be sufficiently small. We apply such method to a vorticity maximization problem for the Navier--Stokes equations and compare with it the augmented Lagrangian method used by C. Dapogny et al. [2].

[1] H. Azegami, K. Takeuchi. A smoothing method for shape optimization: traction method using the Robin condition, Int J Comput Methods. 3(1): 21--33, 2006.

[2] C. Dapogny, P. Frey, F. Omnès, Y. Privat. Geometrical shape optimization in fluid mechanics using FreeFem++, Struct Multidisciplinary Opt 58(6):2761–2788, 2018.

We propose non-conventional shape optimization approaches for the resolution of shape inverse problems inspired by non-destructive testing and evaluation. Our main objective is to improve the detection of the concave parts or regions of the unknown inclusion/obstacle/boundary through two different strategies and under shape optimization settings. Firstly, we will introduce the so-called alternating direction method of multipliers or ADMM in shape optimization framework to solve a boundary inverse problem for the Laplacian with Dirichlet condition using a single boundary measurement. Secondly, we will consider a similar problem, but with the Robin condition, and demonstrate how we can effectively detect a void with concavities using several pairs of Cauchy data. We will illustrate the effectiveness of the proposed schemes by testing them to some shape detection problems with pronounced concavities and under noisy data. Examples are given in two and three dimensions.