Real and Complex Analysis

Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves

Finite element methods for Maxwell's equations.

This review paper provides an overview of different level-set methods for structural topology optimization. Level-set methods can be categorized with respect to the level-set-function parameterization, the geometry mapping, the physical/mechanical model, the information and the procedure to update the design and the applied regularization. Different approaches for each of these interlinked components are outlined and compared. Based on this categorization, the convergence behavior of the optimization process is discussed, as well as control over the slope and smoothness of the level-set function, hole nucleation and the relation of level-set methods to other topology optimization methods. The importance of numerical consistency for understanding and studying the behavior of proposed methods is highlighted. This review concludes with recommendations for future research.

Level-set methods for structural topology optimization: a review

nonlinear functional analysis and its applications is available in our book collection an online access to it is set as public so you can get it instantly. Our books collection hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the nonlinear functional analysis and its applications is universally compatible with any devices to read.

Nonlinear Functional Analysis And Its Applications

In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by some issues arising in population dynamics. The question is to determine optimal spatial arrangements of favorable and unfavorable regions for a species to survive. The mathematical formulation of the model leads to an indefinite weight linear eigenvalue
problem in a fixed box $\Omega$ and we consider the general case of Robin boundary conditions on $\partial\Omega$. It is well known that it suffices to consider {\it bang-bang} weights taking two values of different signs, that can be parametrized by the characteristic function of the subset $E$ of $\Omega$ on which resources are located. Therefore, the optimal spatial arrangement is obtained by minimizing the positive principal eigenvalue with respect to $E$, under a volume constraint. By using symmetrization techniques, as well as necessary optimality conditions, we prove new qualitative results on the solutions. Namely, we completely solve the problem in dimension 1, we prove the counter-intuitive result that the ball is almost never a solution in dimension 2 or higher, despite what suggest the numerical simulations. We also introduce a new rearrangement in the ball allowing to get a better candidate than the ball for optimality when Neumann boundary conditions are imposed. We also provide numerical illustrations of our results and of the optimal configurations.

https://hal.archives-ouvertes.fr/hal-01295457/file/eig_indefinite_revision_hal.pdf

Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions

The goal of this paper is to improve the performance of an electric motor by modifying the geometry of a specific part of the iron core of its rotor. To be more precise, the objective is to smooth the rotation pattern of the rotor. A shape optimization problem is formulated by introducing a tracking-type cost functional to match a desired rotation pattern. The magnetic field generated by permanent magnets is modeled by a nonlinear partial differential equation of magnetostatics. The shape sensitivity analysis is rigorously performed for the nonlinear problem by means of a new shape-Lagrangian formulation adapted to nonlinear problems.

Shape optimization of an electric motor subject to nonlinear magnetostatics

Topological sensitivity analysis is performed for the piecewise constant Mumford-Shah functional. Topological and shape derivatives are combined in order to derive an algorithm for image segmentation with fully automatized initialization. Segmentation of 2D and 3D data is presented. Further, a generalized Mumford-Shah functional is proposed and numerically investigated for the segmentation of images modulated due to, e.g., coil sensitivities.

/pdf/multiphase-image-segmentation-and-modulation-recovery-based-a35gzpbxxj.pdf

Multiphase Image Segmentation and Modulation Recovery Based on Shape and Topological Sensitivity

This paper presents an educational code written using FEniCS, based on the level set method, to perform compliance minimization in structural optimization. We use the concept of distributed shape derivative to compute a descent direction for the compliance, which is defined as a shape functional. The use of the distributed shape derivative is facilitated by FEniCS, which allows to handle complicated partial differential equations with a simple implementation. The code is written in the framework of linearized elasticity, and can be easily adapted to tackle other functionals and partial differential equations. We also provide an extension of the code for compliant mechanisms. We start by explaining how to compute shape derivatives, and briefly discuss the differences between the distributed and boundary expressions of the shape derivative. Then we describe the implementation in details, and show the application of this code to some classical benchmarks of topology optimization. The files are provided in Online Resource 1 to 3 and also available at http://antoinelaurain.com/compliance.htm
 . The main file is also given in the Appendix.

A level set-based structural optimization code using FEniCS

The shape of the free boundary arising from the solution of a variational inequality is controlled by the shape of the domain where the variational inequality is defined. Shape and topological sensitivity analysis is performed for the obstacle problem and for a regularized version of its primal-dual formulation. The shape derivative for the regularized problem can be defined and converges to the solution of a linear problem. These results are applied to an inverse problem and to the electrochemical machining problem.

/pdf/optimal-shape-design-subject-to-elliptic-variational-4dt6j7bgps.pdf

Antoine Laurain

Papers

Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions

Shape optimization of an electric motor subject to nonlinear magnetostatics

Multiphase Image Segmentation and Modulation Recovery Based on Shape and Topological Sensitivity

A level set-based structural optimization code using FEniCS

Optimal Shape Design Subject to Elliptic Variational Inequalities