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**Finite element handbook.**
*(English)*
Zbl 0638.65076

New York etc.: McGraw-Hill Book Company. XXIV, 1360, I-20 p.; DM 237.55 (1987).

An encyclopedic work which is too voluminous to be held in hand is usually called “Handbook”. The purpose of the Handbook under review is, as stated in the Preface, to present the underlying mathematical principles, the fundamental formulations, and both commonly used and specialized applications of the finite element method (FEM) in a single volume. The book consists of four parts with the headings “FEM mathematics”, “FEM fundamentals”, “FEM applications”, and “FEM computations”, which are addressed, loosely speaking, to pure mathematicians, applied mathematicians, engineers, and computer specialists, respectively.

Part 1 of the Handbook begins by considering functional analysis, functional spaces, and partial differential equations, to establish the mathematical foundations for finite-element methods. Then follows the concise and elegant theory for affine-equivalent and almost-affine- equivalent finite elements. This theory has proved to be a highly valuable tool for numerical analysis in such areas as isoparametric and curved elements, singular and rational elements, composite elements, numerical integration, and nonconforming methods. The succeeding chapters deal with the finite-element methods for elliptic boundary-value and eigenvalue problems and for time-dependent problems. In the last chapter, finite-element methods for variational inequalities are examined.

In Part 2, variational principles and constitutive equations are first reviewed. Then, the fundamental steps in the variational finite-element method are presented - from initial discretization and choice of element through solution of the system matrix equation. This is followed by detailed treatments of finite elements based on displacement fields and by coverage of mixed and hybrid finite-element methods. Discussion then moves through other types of finite element method to a review of finite- element methods for instability analysis and transient-response analysis. The final chapter brings together important information on error estimates, convergence ratios, and stability.

Part 3 contains a wide-ranging consideration of finite-element applications within solid mechanics, fluid mechanics, geomechanics, aeromechanics, biomechanics, chemical reactions, nuclear reactors, plasmas, acoustics, and electromagnetics. Coupled systems variously involving fluids, structures, and soils are also studied. The last and very important chapter assesses errors in finite-element computation and considers the principles for selecting a mesh-design strategy.

Part 4 begins by reviewing techniques for solving the system matrix equations of finite-element analysis, and then considers reanalysis, nodal synthesis, static condensation, and substructuring. Next follow topics in modeling, and pre- and postprocessing. New directions in computer technology in the recent past and near future are then assessed and related to the anticipated development of finite-element hardware and software systems. The final chapter presents an overview of the current capabilities of finite-element software packages.

Collecting the contributions of 96 authors from various scientific or industrial institutions and with completely different viewpoints, within a single volume, is undoubtedly a highly nontrivial and risky project. In the present case, the editors succeeded certainly in not only putting together the most comprehensive survey of finite element theory and practice available at all, but also presenting this survey in a coherent and homogeneous manner. The presentation is high-standard, self- consistent, and very clear throughout, as may be seen, for instance, already in the first three chapters on functional analysis and partial differential equations (written by F. Brezzi and G. Gilardi) which contains much general information and lots of illuminating examples.

Without any doubt, this Handbook will become a standard reference and valuable source for all researchers in numerical mathematics, engineering, and technology.

Part 1 of the Handbook begins by considering functional analysis, functional spaces, and partial differential equations, to establish the mathematical foundations for finite-element methods. Then follows the concise and elegant theory for affine-equivalent and almost-affine- equivalent finite elements. This theory has proved to be a highly valuable tool for numerical analysis in such areas as isoparametric and curved elements, singular and rational elements, composite elements, numerical integration, and nonconforming methods. The succeeding chapters deal with the finite-element methods for elliptic boundary-value and eigenvalue problems and for time-dependent problems. In the last chapter, finite-element methods for variational inequalities are examined.

In Part 2, variational principles and constitutive equations are first reviewed. Then, the fundamental steps in the variational finite-element method are presented - from initial discretization and choice of element through solution of the system matrix equation. This is followed by detailed treatments of finite elements based on displacement fields and by coverage of mixed and hybrid finite-element methods. Discussion then moves through other types of finite element method to a review of finite- element methods for instability analysis and transient-response analysis. The final chapter brings together important information on error estimates, convergence ratios, and stability.

Part 3 contains a wide-ranging consideration of finite-element applications within solid mechanics, fluid mechanics, geomechanics, aeromechanics, biomechanics, chemical reactions, nuclear reactors, plasmas, acoustics, and electromagnetics. Coupled systems variously involving fluids, structures, and soils are also studied. The last and very important chapter assesses errors in finite-element computation and considers the principles for selecting a mesh-design strategy.

Part 4 begins by reviewing techniques for solving the system matrix equations of finite-element analysis, and then considers reanalysis, nodal synthesis, static condensation, and substructuring. Next follow topics in modeling, and pre- and postprocessing. New directions in computer technology in the recent past and near future are then assessed and related to the anticipated development of finite-element hardware and software systems. The final chapter presents an overview of the current capabilities of finite-element software packages.

Collecting the contributions of 96 authors from various scientific or industrial institutions and with completely different viewpoints, within a single volume, is undoubtedly a highly nontrivial and risky project. In the present case, the editors succeeded certainly in not only putting together the most comprehensive survey of finite element theory and practice available at all, but also presenting this survey in a coherent and homogeneous manner. The presentation is high-standard, self- consistent, and very clear throughout, as may be seen, for instance, already in the first three chapters on functional analysis and partial differential equations (written by F. Brezzi and G. Gilardi) which contains much general information and lots of illuminating examples.

Without any doubt, this Handbook will become a standard reference and valuable source for all researchers in numerical mathematics, engineering, and technology.

Reviewer: J.Appell

### MSC:

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

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

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

00A20 | Dictionaries and other general reference works |

00Bxx | Conference proceedings and collections of articles |

46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |

65D15 | Algorithms for approximation of functions |

74S99 | Numerical and other methods in solid mechanics |

76M99 | Basic methods in fluid mechanics |

86-08 | Computational methods for problems pertaining to geophysics |

92-08 | Computational methods for problems pertaining to biology |

80M99 | Basic methods in thermodynamics and heat transfer |

82-08 | Computational methods (statistical mechanics) (MSC2010) |

65Fxx | Numerical linear algebra |

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

49J40 | Variational inequalities |