Non-local control in the conduction coefficients: well posedness and convergence to the local limit

We consider a problem of optimal distribution of conductivities in a system governed by a non-local diffusion law. The problem stems from applications in optimal design and more specifically topology optimization. We propose a novel parametrization of non-local material properties. With this parametrization the non-local diffusion law in the limit of vanishing non-local interaction horizons converges to the famous and ubiquitously used generalized Laplacian with SIMP (Solid Isotropic Material with Penalization) material model. The optimal control problem for the limiting local model is typically ill-posed and does not attain its infimum without additional regularization. Surprisingly, its non-local counterpart attains its global minima in many practical situations, as we demonstrate in this work. In spite of this qualitatively different behaviour, we are able to partially characterize the relationship between the non-local and the local optimal control problems. We also complement our theoretical findings with numerical examples, which illustrate the viability of our approach to optimal design practitioners

Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients

We present a convergence analysis of a cell-based finite volume (FV) discretization scheme applied to a problem of control in the coefficients of a generalized Laplace equation modelling, for example, a steady state heat conduction. Such problems arise in applications dealing with geometric optimal design, in particular shape and topology optimization, and are most often solved numerically utilizing a finite element approach. Within the FV framework for control in the coefficients problems the main difficulty we face is the need to analyze the convergence of fluxes defined on the faces of cells, whereas the convergence of the coefficients happens only with respect to the “volumetric” Lebesgue measure. Additionally, depending on whether the stationarity conditions are stated for the discretized or the original continuous problem, two distinct concepts of stationarity at a discrete level arise. We provide characterizations of limit points, with respect to FV mesh size, of globally optimal solutions and two types of stationary points to the discretized problems. We illustrate the practical behaviour of our cell-based FV discretization algorithm on a numerical example

The dual approach to optimal control in the coefficients of nonlocal nonlinear diffusion

We derive the dual variational principle (principle of minimal complementary energy) for the nonlocal nonlinear scalar diffusion problem, which may be viewed as the nonlocal version of the $p$-Laplacian operator. We establish existence and uniqueness of solutions (two-point fluxes) as well as their quantitative stability, which holds uniformly with respect to the small parameter (nonlocal horizon) characterizing the nonlocality of the problem. We then focus on the nonlocal analogue of the classical optimal control in the coefficient problem associated with the dual variational principle, which may be interpreted as that of optimally distributing a limited amount of conductivity in order to minimize the complementary energy. We show that this nonlocal optimal control problem $\Gamma$-converges to its local counterpart, when the nonlocal horizon vanishes.Comment: 37 pages, 0 figure

From nonlocal Eringen's model to fractional elasticity

Eringen's model is one of the most popular theories in nonlocal elasticity. It has been applied to many practical situations with the objective of removing the anomalous stress concentrations around geometric shape singularities, which appear when the local modelling is used. Despite the great popularity of Eringen's model in mechanical engineering community, even the most basic questions such as the existence and uniqueness of solutions have been rarely considered in the research literature for this model. In this work we focus on precisely these questions, proving that the model is in general ill-posed in the case of smooth kernels, the case which appears rather often in numerical studies. We also consider the case of singular, non-smooth kernels, and for the paradigmatic case of the Riesz potential we establish the well-posedness of the model in fractional Sobolev spaces. For such a kernel, in dimension one the model reduces to the well-known fractional Laplacian. Finally, we discuss possible extensions of Eringen's model to spatially heterogeneous material distributions

Application of a Parametric Level-Set Approach to Topology Optimization of Fluids with the Navier–Stokes and Lattice Boltzmann Equations

Traditional material distribution based methods applied to the topology optimization of fluidic systems often suffer from rather slow convergence. The local influence of the design variables in the traditional material distribution based approaches is seen as the primary cause, leading to small gradients which cannot drive the optimization process sufficiently. The present work is an attempt to improve the rate of convergence of topology optimization methods of fluidic systems by employing a parametric level-set function coupled with a topology description approach. Using level-set methods, a global impact of design variables is achieved and the material description is decoupled from the flow field discretization. This promises to improve the gradients with respect to the design variables and can be applied to rather different types of fluid formulations and discretization methods. In the present work, a finite element method for solving the Navier-Stokes equations and a hydrodynamic finite difference lattice Boltzmann method are considered. Using a 2D example the parametric level-set approach is validated through comparison with traditional material distribution based methods. While the parametric level-set approach leads to the desired optimal designs and has advantages such as improved modularity and smoothness of design boundaries when compared to material distribution based methods, the present study does not reveal improvements for the convergence of the optimization problem