The finite element analysis of shells – fundamentals.

*(English)*Zbl 1103.74003
Computational Fluid and Solid Mechanics. Berlin: Springer (ISBN 3-540-41339-1/hbk). xi, 330 p. (2003).

The main objective is to combine the fundamentals of shell theories, mathematical analyses and finite element methods in order to develop shell finite element approximations which are reliable and effective. With such finite element methods, engineers can confidently replace a great part of expensive laboratory experiments by really efficient numerical simulations.

Chapter 1 introduces shell structures in engineering designs and gives a short description of objectives of the book. Basic results from differential geometry required for further developments are detailed in Chapter 2. Likewise, elements of functional and numerical analysis are recalled in Chapter 3, with special attention to stability and convergence properties of abstract finite element discretizations, in particular mixed formulations. Next, Chapter 4 discusses the “linear basic shell model” from which other classical shell and plate models can be derived, and it introduces a proper mathematical framework which allows to prove the corresponding existence and uniqueness of solutions for each of them. The basic characteristic of a shell structure is that one of its dimensions, the thickness \(t\), is small with respect to the two others. Chapter 5 studies the asymptotic behavior of shell models in order to know whether the model converges towards a limit model when \(t\) tends to zero.

Chapter 6 describes and analyzes the displacement-based finite element procedures, i.e. the finite element approximations directly obtained by applying the variational principle in the finite element space which discretizes the space of admissible displacements for the structure. These studies include not only the direct discretization of shell models, but also the facet-shell element approach and the general shell element approach.

Chapter 7 describes and analyzes numerical locking in thin structures, the crucial problem to develop reliable and effective finite element procedures. Indeed, it was soon observed that finite element approximations tend to dramatically deteriorate when the thickness \(t\) of the structure decreases, a phenomenon which is directly associated with a possible nonuniform convergence with respect to \(t\) of two-dimensional finite element approximations. Examples of such numerical lockings are presented, their treatments by mixed formulations are analyzed in the framework of thin plate problems, and specific difficulties arising in the analysis of shells are discussed. The objective of Chapter 8 is to propose some strategies to evaluate shell finite element discretizations in the search for improved approximation algorithms. First, guidelines for assessing and improving the reliability of shell finite elements are presented and second, formulation of MITC shell elements and some assessment results are discussed.

All previous developments concern the linear analysis of shells. The aim of Chapter 9 is to underline that some of these results can be extended to the nonlinear analysis of shells as well, through incremental algorithms.

The expertises of both authors combined with a very clear and attractive presentation lead to a wonderful masterpiece which is, with no doubt and for a long time, the great reference book for the development of accurate and reliable numerical methods in the thin shell world.

Chapter 1 introduces shell structures in engineering designs and gives a short description of objectives of the book. Basic results from differential geometry required for further developments are detailed in Chapter 2. Likewise, elements of functional and numerical analysis are recalled in Chapter 3, with special attention to stability and convergence properties of abstract finite element discretizations, in particular mixed formulations. Next, Chapter 4 discusses the “linear basic shell model” from which other classical shell and plate models can be derived, and it introduces a proper mathematical framework which allows to prove the corresponding existence and uniqueness of solutions for each of them. The basic characteristic of a shell structure is that one of its dimensions, the thickness \(t\), is small with respect to the two others. Chapter 5 studies the asymptotic behavior of shell models in order to know whether the model converges towards a limit model when \(t\) tends to zero.

Chapter 6 describes and analyzes the displacement-based finite element procedures, i.e. the finite element approximations directly obtained by applying the variational principle in the finite element space which discretizes the space of admissible displacements for the structure. These studies include not only the direct discretization of shell models, but also the facet-shell element approach and the general shell element approach.

Chapter 7 describes and analyzes numerical locking in thin structures, the crucial problem to develop reliable and effective finite element procedures. Indeed, it was soon observed that finite element approximations tend to dramatically deteriorate when the thickness \(t\) of the structure decreases, a phenomenon which is directly associated with a possible nonuniform convergence with respect to \(t\) of two-dimensional finite element approximations. Examples of such numerical lockings are presented, their treatments by mixed formulations are analyzed in the framework of thin plate problems, and specific difficulties arising in the analysis of shells are discussed. The objective of Chapter 8 is to propose some strategies to evaluate shell finite element discretizations in the search for improved approximation algorithms. First, guidelines for assessing and improving the reliability of shell finite elements are presented and second, formulation of MITC shell elements and some assessment results are discussed.

All previous developments concern the linear analysis of shells. The aim of Chapter 9 is to underline that some of these results can be extended to the nonlinear analysis of shells as well, through incremental algorithms.

The expertises of both authors combined with a very clear and attractive presentation lead to a wonderful masterpiece which is, with no doubt and for a long time, the great reference book for the development of accurate and reliable numerical methods in the thin shell world.

Reviewer: Michel Bernadou (Paris La Defense)

##### MSC:

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74S05 | Finite element methods applied to problems in solid mechanics |

74K25 | Shells |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N15 | Error bounds for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |