Contents -- Preface -- Philippe G. Ciarlet: An Introduction to Differential Geometry in R3 -- Introduction -- 1 Three-dimensional differential geometry Outline -- 1.1 Curvilinear coordinates -- 1.2 Metric tensor -- 1.3 Volumes, areas, and lengths in curvilinear coordinates -- 1.4 Covariant derivatives of a vector field -- 1.5 Necessary conditions satisfied by the metric tensor -- the Riemann curvature tensor -- 1.6 Existence of an immersion defined on an open set in R3 with a prescribed metric tensor -- 1.7 Uniqueness up to isometries of immersions with the same metric tensor -- 1.8 Continuity of an immersion as a function of its metric tensor -- 2 Differential geometry of surfaces Out line -- 2.1 Curvilinear coordinates on a surface -- 2.2 First fundamental form -- 2.3 Areas and lengths on a surface -- 2.4 Second fundamental form -- curvature on a surface -- 2.5 Principal curvatures -- Gaussian curvature -- 2.6 Covariant derivatives of a vector field defined on a surface -- the Gauss and Weingarten formulas -- 2.7 Necessary conditions satisfied by the first and second fundamental forms: the Gauss and Codazzi-Mainardi equations -- Gauss' Theorema Ggregium -- 2.8 Existence of a surface with prescribed first and second fundamental forms -- 2.9 Uniqueness up to proper isometries of surfaces with the same fundamental forms -- 2.10 Continuity of a surface as a function of its fundamental forms -- References -- Philzppe G. Ciarlet, Cristinel Mardare: An Introduction to Shell Theory -- Introduction -- 1 Three-dimensional theory Outline -- 1.1 Notation, definitions, and some basic formulas -- 1.2 Equations of equilibrium -- 1.3 Constitutive equations of elastic materials -- 1.4 The equations of nonlinear and linearized three-dimensional elasticity -- 1.5 A fundamental lemma of J.L. Lions -- 1.6 Existence theory in linearized three-dimensional elasticity.

1.7 Existence theory in nonlinear three-dimensional elasticity by the implicit function theorem -- 1.8 Existence theory in nonlinear three-dimensional elasticity by the minimization of energy (John Ball's approach) -- 2 Two-dimensional theory Outline -- 2.1 A quick review of the differential geometry of surfaces in R3 -- 2.2 Geometry of a shell -- 2.3 The three-dimensional shell equations -- 2.4 The two-dimensional approach to shell theory -- 2.5 Nonlinear shell models obtained by r-convergence -- 2.6 Linear shell models obtained by asymptotic analysis -- 2.7 The nonlinear Koiter shell model -- 2.8 The linear Koiter shell model -- 2.9 Korn's inequalities on a surface -- 2.10 Existence, uniqueness, and regularity of the solution to the linear Koiter shell model -- References -- Dominique Chapelle: Some New Results and Current Challenges in the Finite Element Analysis of Shells -- 1 Introduction -- 2 Two families of shell finite elements -- 2.1 Discretizations of classical shell models -- 2.2 General shell elements -- 3 Computational reliability issues for thin shells -- 3.1 Asymptotic behaviours of shell models -- 3.2 Asymptotic reliability of shell finite elements -- 3.3 Guidelines for assessing and improving the reli- ability of shell finite elements -- Acknowledgement -- References -- Pascal Frey: A Differential Geometry Approach to Mesh Generation -- Introduction -- 1 Preliminary definitions -- 1.1 Triangulations and meshes -- 1.1.1 Numerical simulations -- 1.1.2 A brief survey of mesh generation methods -- 1.1.3 Mesh adaptation -- 1.2 Notion of metric tensor -- 1.2.1 Metric, scalar product and distance -- 1.2.2 Metric decomposition -- 1.2.3 Geometric representation -- 1.2.4 Metric intersection -- 1.2.5 Metric interpolation -- 1.3 A differential geometry primer -- 1.3.1 The first fundamental form -- area -- 1.3.2 The second fundamental form.

curvatures -- 2 A geometric error estimate -- 2.1 A priori error analysis -- 2.1.1 Interpolation error in L2 norm and H1 seminorm -- 2.1.2 PI elements in one dimension -- 2.1.3 Lagrange PI elements in two dimensions -- 2.2 Mesh adaptation using the error estimate -- 2.2.1 Anisotropic formulation of the interpolation error -- 2.2.2 Problem statement -- 2.2.3 Metric construction -- 2.2.4 Evaluation of the Hessian matrix -- 2.2.5 An error estimate for CFD problems -- 3 Mesh adaptation using a geometric error estimate -- 3.1 Classical vs. adaptive Delaunay mesh generation -- 3.1.1 The Delaunay triangulation -- 3.1.2 Constrained triangulation -- 3.1.3 Delaunay-based meshing algorithm -- 3.1.4 Creation and insertion of internal points -- 3.2 Surface mesh generation -- 3.2.1 Problem statement: the concept of geometric mesh -- 3.2.2 Local deformation of a surface -- 3.2.3 Local curvature of a surface -- 3.2.4 Evaluation of the intrinsic properties of a discrete sur- face -- 4 Applications -- 4.1 An academic example -- 4.2 Curvature-driven evolution flows -- 4.3 A CFD example in two dimensions -- 4.4 Curvature-based surface mesh adaptation -- 4.5 Mesh adaptation in three dimensions -- 5 Conclusions -- Acknowledgments -- References.