There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

A Posteriori Error Estimation in Finite Element Analysis

Convex Analysis: (pms-28)

We discuss recent advances on robust unfitted finite element methods on cut meshes. These methods are designed to facilitate computations on complex geometries obtained, for example, from computer- ...

CutFEM: Discretizing geometry and partial differential equations

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/131866/1/j.cma.2010.05.013.pdf

A topology optimization method based on the level set method incorporating a fictitious interface energy

Etude des problemes unilateraux pour les fonctions scalaires. Contact unilateral de corps elastiques. Problemes de la theorie de la plasticite

Solution of Variational Inequalities in Mechanics

Numerical methods for unilateral problems in solid mechanics

Preliminaries. Abstract Setting of the Optimal Shape Design Problem and Its Approximation. Optimal Shape Design of Systems Governed by a Unilateral Boundary Value State Problem the Scalar Case. Approximation of the Optimal Shape Design Problems by Finite Elements the Scalar Case. Numerical Realization of Optimal Shape Design Problems Associated with a Unilateral Boundary Value Problem the Scalar Case. Shape optimization in Unilateral Boundary Value Problems with a "Flux" Cost Functional. Optimal Shape Design Contact Problems the Elastic Case. Shape Optimization of Materially Non-linear Bodies in Contact. Shape Optimization in Problems with Inner Obstacles. Optimum Composite Material Design. Topology Optimization in Unilateral Problems. Appendices. Bibliography. Index.

Finite Element Approximation for Optimal Shape, Material and Topology Design

Preface. List of Notations. Introduction. Part I: Introductory Topics. 1. Mathematical Preliminaries. 2. Nonsmooth Mechanics. Part II: Finite Element Approximation of Hemivariational Inequalities. 3. Approximation of Elliptic Hemivariational Inequalities. 4. Time Dependent Case. Part III: Nonsmooth Optimization Methods. 5. Nonsmooth Optimization Methods. Part IV: Numerical Examples. 6. Numerical Examples. Index.

Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications

The purpose of this paper is to present a new fictitious domain approach inspired by the extended finite element method introduced by Moes, Dolbow, and Belytschko in [Internat. J. Numer. Methods Engrg., 46 (1999), pp. 131-150]. An optimal method is obtained thanks to an additional stabilization technique. Some a priori estimates are established and numerical experiments illustrate different aspects of the method. The presentation is made on a simple Poisson problem with mixed Neumann and Dirichlet boundary conditions. The extension to other problems or boundary conditions is quite straightforward.

/pdf/a-new-fictitious-domain-approach-inspired-by-the-extended-17qqy8qmqd.pdf

A New Fictitious Domain Approach Inspired by the Extended Finite Element Method

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