Alberto Paganini

Shape Optimization by Pursuing Diffeomorphisms

Abstract We consider PDE constrained shape optimization in the framework of finite element discretization of the underlying boundary value problem. We present an algorithm tailored to preserve and exploit the approximation properties of the finite element method, and that allows for arbitrarily high resolution of shapes. It employs (i) B-spline based representations of the deformation diffeomorphism, and (ii) superconvergent domain integral expressions for the shape gradient. We provide numerical evidence of the performance of this method both on prototypical well-posed and ill-posed shape optimization problems.

Approximate Shape Gradients for Interface Problems

Shape gradients of shape differentiable shape functionals constrained to an interface problem (IP) can be formulated in two equivalent ways. Both formulations rely on the solution of two IPs, and their equivalence breaks down when these IPs are solved approximatively. We establish which expression for the shape gradient offers better accuracy for approximations by means of finite elements. Great effort is devoted to provide numerical evidence of the theoretical considerations.

Fast convolution quadrature based impedance boundary conditions

We consider an eddy current problem in time-domain relying on impedance boundary conditions on the surface of the conductor(s). We pursue its full discretization comprising (i) a finite element Galerkin discretization by means of lowest order edge elements in space, and (ii) temporal discretization based on Runge-Kutta Convolution Quadrature (CQ) for the resulting Volterra integral equation in time. The final algorithm also involves the fast and oblivious approximation of CQ. For this method we give a comprehensive convergence analysis and establish that the errors of spatial discretization, CQ and of its approximate realization add up to the final error bound.

Shape optimization of microlenses

Microlenses are highly attractive for optical applications such as super resolution and photonic nanojets, but their design is more demanding than the one of larger lenses because resonance effects play an important role, which forces one to perform a full wave analysis. Although mostly spherical microlenses were studied in the past, they may have various shapes and their optimization is highly demanding, especially, when the shape is described with many parameters. We first outline a very powerful mathematical tool: shape optimization based on shape gradient computations. This procedure may be applied with much less numerical cost than traditional optimizers, especially when the number of parameters describing the shape goes to infinity. In order to demonstrate the concept, we optimize microlenses using shape optimization starting from more or less reasonable elliptical and semi-circular shapes. We show that strong increases of the performance of the lenses may be obtained for any reasonable value of the refraction index.

Formal Solutions for Polarized Radiative Transfer. III. Stiffness and Instability

Efficient numerical approximation of the polarized radiative transfer equation is challenging because this system of ordinary differential equations exhibits stiff behavior, which potentially results in numerical instability. This negatively impacts the accuracy of formal solvers, and small step-sizes are often necessary to retrieve physical solutions. This work presents stability analyses of formal solvers for the radiative transfer equation of polarized light, identifies instability issues, and suggests practical remedies. In particular, the assumptions and the limitations of the stability analysis of Runge-Kutta methods play a crucial role. On this basis, a suitable and pragmatic formal solver is outlined and tested. An insightful comparison to the scalar radiative transfer equation is also presented.

Trefftz Approximations: A New Framework for Nonreflecting Boundary Conditions

A new general framework for approximate nonreflecting boundary conditions in wave scattering involves a set of local Trefftz functions-outgoing waves-and a commensurate set of degrees of freedom (DOF). With specific choices of bases and DOF, one obtains classical Engquist-Majda and Bayliss-Turkel conditions. Other choices yield a variety of approximate conditions. With derivatives of the solution as additional boundary DOF, accurate results can be obtained even on fairly coarse grids, as illustrated numerically.

Approximate Riesz Representatives of Shape Gradients

We study finite element approximations of Riesz representatives of shape gradients. First, we provide a general perspective on its error analysis. Then, we focus on shape functionals constrained by elliptic boundary value problems and \(H^1\)-representatives of shape gradients. We prove linear convergence in the energy norm for linear Lagrangian finite element approximations. This theoretical result is confirmed by several numerical experiments.

Efficient convolution based impedance boundary conditions

Purpose – The purpose of this paper is to propose an approach based on Convolution quadrature (CQ) for the modeling and the numerical treatment of impedance boundary condition. Design/methodology/approach – The model is derived from a general setting. Its discretization is discussed in details by providing pseudo-codes and by performing their complexity analysis. The model is validated through several numerical experiments. Findings – CQ provides an efficient and accurate treatment of impedance boundary conditions. Originality/value – The paper suggests a new effective treatment of impedance boundary conditions.