Differential Geometry: Theory and Applications

The following sections are included:

• Preface

• Contents

The following sections are included:

• Introduction

• Three-dimensional differential geometry
  • Curvilinear coordinates
  • Metric tensor
  • Volumes, areas, and lengths in curvilinear coordinates
  • Covariant derivatives of a vector field
  • Necessary conditions satisfied by the metric tensor; the Riemann curvature tensor
  • Existence of an immersion defined on an open set in ℝ³ with a prescribed metric tensor
  • Uniqueness up to isometries of immersions with the same metric tensor
  • Continuity of an immersion as a function of its metric tensor

• Differential geometry of surfaces
  • Curvilinear coordinates on a surface
  • First fundamental form
  • Areas and lengths on a surface
  • Second fundamental form; curvature on a surface
  • Principal curvatures; Gaussian curvature
  • Covariant derivatives of a vector field defined on a surface; the Gauss and Weingarten formulas
  • Necessary conditions satisfied by the first and second fundamental forms: the Gauss and Codazzi-Mainardi equations; Gauss' Theorema Ggregium
  • Existence of a surface with prescribed first and second fundamental forms
  • Uniqueness up to proper isometries of surfaces with the same fundamental forms
  • Continuity of a surface as a function of its fundamental forms

• References

The following sections are included:

• Introduction

• Three-dimensional theory
  • Notation, definitions, and some basic formulas
  • Equations of equilibrium
  • Constitutive equations of elastic materials
  • The equations of nonlinear and linearized three-dimensional elasticity
  • A fundamental lemma of J.L. Lions
  • Existence theory in linearized three-dimensional elasticity
  • Existence theory in nonlinear three-dimensional elasticity by the implicit function theorem
  • Existence theory in nonlinear three-dimensional elasticity by the minimization of energy (John Ball's approach)

• Two-dimensional theory
  • A quick review of the differential geometry of surfaces in ℝ³
  • Geometry of a shell
  • The three-dimensional shell equations
  • The two-dimensional approach to shell theory
  • Nonlinear shell models obtained by Γ-convergence
  • Linear shell models obtained by asymptotic analysis
  • The nonlinear Koiter shell model
  • The linear Koiter shell model
  • Korn's inequalities on a surface
  • Existence, uniqueness, and regularity of the solution to the linear Koiter shell model

• References

Finite element procedures are widely used in engineering practice to analyse the behaviour of shell structures. However, the concept of "shell finite element" is still somewhat fuzzy, as it may correspond to very different ideas and techniques in various actual implementations. In particular, a significant distinction can be made between shell elements that are obtained via the discretization of shell models, and shell elements—such as the general shell elements—derived from 3D formulations using some kinematical assumptions, without the use of any shell theory. Our first objective in this paper is to give a unified perspective of these two families of shell elements. This is expected to be very useful as it paves the way for further thorough mathematical analyses of shell elements. A particularly important motivation for this is the understanding and the treatment of the deficiencies associated with the analysis of thin shells (among which the locking phenomenon). We then survey these deficiencies, in the framework of the asymptotic behaviours of shell models. We conclude the article by giving some detailed guidelines to numerically assess the performance of shell finite elements as regards these pathological phenomena, which is essential to design improved procedures.

In this chapter, we present a mesh adaptation scheme based on a geometric error estimate. The term geometric refers here to the manner we are considering the numerical solution of a partial derivative equation. Indeed, the solution is envisaged as a manifold embedded in a higher dimensional space, for instance a two-dimensional Cartesian surface embedded in ℝ³. This allows to study the variations of the solution in terms of the calculus of variations and more precisely to take advantage of the powerful tool of differential geometry to properly characterize the surface.

Theoretical issues about error estimate in H¹ seminorm or in L^p norms are discussed, the main concepts of mesh generation and adaptation are briefly presented and practical issues related to the implementation of the numerical algorithms are discussed. Numerical examples from various fields of applications are presented to emphasize the potential of this approach.