The finite element method. Linear static and dynamic finite element analysis.

*(English)*Zbl 0634.73056
Englewood Cliffs, New Jersey: Prentice-Hall International, Inc. XXVII, 803 p.; £17.95; $ 28.95 (1987).

The author uses the weighted-residual model - the weak form of the boundary value problems solved by means of Galerkin’s approximation method - to obtain the established numerical schemes of the FEM as they are known by engineers. This is an undeniable achievement of the book under review, for at least three reasons: (1) it provides a rigorous mathematical framework, based on the principles and theorems of functional analysis, for formulating, explaining, understanding, and developing the FEM; (2) it allows for the extending of applications of the FEM to the solution of problems for which no energy functional can be defined explicitly (such as those generated by the Navier-Stokes equation), avoiding the so-called “variational crimes” in the standard approach to these problems; (3) it makes it possible to define mixed and hybrid finite elements (to this end one can also employ the Hellinger- Reissner or Hu-Washizu generalized variational principles).

In chapter 1, the author defines and formulates one-dimensional boundary- value problems, introduces the concept of meshes - finite element discretization - and expresses the element matrices and vectors as well as the numerical procedures for the assembly of finite elements into global matrices and vectors, and the ways they are to be stored in computer memory (the necessary arrays). Chapter 2 discusses some two- dimensional and three-dimensional boundary-value problems.

Chapter 3 deals with finite element approximation, defining simplex bilinear finite elements and interpolation functions, isoparametric finite elements, and the finite elements with variable numbers of nodes. The author presents the algorithm of the Gauss method for the numerical integration of the expressions of the stiffness matrices and the force vectors, by means of which the various types of finite elements are characterized.

Chapter 4 begins with an analysis of the accuracy of finite element approximations. Starting from the limits of the FEM in the standard (variational) formulation, the author defines the mixed finite elements and the hybrid finite elements. The models involving mixed finite elements are treated as approximations of problems involving constraints, in which the overall energy is no longer convex. Such a situation may be due to the use of Lagrange multipliers - the approach used in this book - when the functional is convex relative to a variable and concave relative to another variable. Hybrid models represent particular cases of mixed models, in which constraints are associated with continuity conditions at the boundary between neighbouring finite elements. Mixed finite elements are defined not only by means of the Lagrange multiplier method, but also by means of an approximate version of this method - the penalty method -, which is similar, in principle, to the way boundary conditions are introduced with a view to solving the global equation system in the FEM. The author also dwells on reduced and selective numerical integration techniques and, as an extension of these techniques, the mean-dilatation approach for anisotropic media. The chapter closes with a treatment of nonconforming finite elements.

In chapter 5 the author applies the Reissner-Mindlin model to the study of plane plates and straight-beam elastic frameworks and defines the finite elements characterized by C(sup 0) interpolation functions. Chapter 6 extends the application of \(C^ 0\) interpolation functions to the definitions of certain curved finite elements.

Chapter 7 marks the beginning of the second part of the book, the one which deals with dynamic analysis problems. This chapter dwells on parabolic, hyperbolic and elliptic eigenvalue problems and their applications in elastostatics, heat conduction, viscous fluid flow, the study of variations, and stability analysis. In chapter 8 the author puts forth algorithms for solving first-order nonstationary problems - parabolic problems. The Euler and Crank-Nicolson methods are presented, and the convergence and stability of the solutions are improved. Algorithms are presented for solving this category of problems by means of element-by-element methods.

Chapter 9 includes algorithms for solving second-order nonstationary problems - hyperbolic and parabolic-hyperbolic problems. The author describes and compares the Houbolt, Newmark, Park, Wilson and predictor- corrector approaches. Chapter 10 discusses numerical procedures for solving eigenvector and eigenvalue problems: the Jacobi method, subspace iteration, the Lanczos algorithm, the Rayleigh-Ritz and Irons-Guyan reduction techniques, and static condensation. The author also describes a package of subroutines written in FORTRAN 77 and based on the Lanczos method.

Chapter 11 presents the computer program DLEARN, which is written in FORTRAN 77 and implements the theoretical and numerical concepts introduced in chapters 3, 5, 7, 9, and 10. The program is intended to deal with linear static and dynamic analysis problems and involves two types of finite elements - the four-node quadrilateral, elastic continuum element and the three-dimensional, elastic truss element. The structures of the program is modular and multipurpose, its open-ended architecture enabling the user to extend and/or alter the program.

The book is undoubtedly original as far as both overall presentation and the content of certain chapters (particularly chapter 4, but also chapters 5, 6, 10, and 11) are concerned. It can safely be concluded that this book goes beyond its avowed purpose - that of providing introductory material to the students who want to study the FEM - and is a valuable addition to the list of reference works covering this field, since the author not only presents the method in question, but also comes up with new developments.

In chapter 1, the author defines and formulates one-dimensional boundary- value problems, introduces the concept of meshes - finite element discretization - and expresses the element matrices and vectors as well as the numerical procedures for the assembly of finite elements into global matrices and vectors, and the ways they are to be stored in computer memory (the necessary arrays). Chapter 2 discusses some two- dimensional and three-dimensional boundary-value problems.

Chapter 3 deals with finite element approximation, defining simplex bilinear finite elements and interpolation functions, isoparametric finite elements, and the finite elements with variable numbers of nodes. The author presents the algorithm of the Gauss method for the numerical integration of the expressions of the stiffness matrices and the force vectors, by means of which the various types of finite elements are characterized.

Chapter 4 begins with an analysis of the accuracy of finite element approximations. Starting from the limits of the FEM in the standard (variational) formulation, the author defines the mixed finite elements and the hybrid finite elements. The models involving mixed finite elements are treated as approximations of problems involving constraints, in which the overall energy is no longer convex. Such a situation may be due to the use of Lagrange multipliers - the approach used in this book - when the functional is convex relative to a variable and concave relative to another variable. Hybrid models represent particular cases of mixed models, in which constraints are associated with continuity conditions at the boundary between neighbouring finite elements. Mixed finite elements are defined not only by means of the Lagrange multiplier method, but also by means of an approximate version of this method - the penalty method -, which is similar, in principle, to the way boundary conditions are introduced with a view to solving the global equation system in the FEM. The author also dwells on reduced and selective numerical integration techniques and, as an extension of these techniques, the mean-dilatation approach for anisotropic media. The chapter closes with a treatment of nonconforming finite elements.

In chapter 5 the author applies the Reissner-Mindlin model to the study of plane plates and straight-beam elastic frameworks and defines the finite elements characterized by C(sup 0) interpolation functions. Chapter 6 extends the application of \(C^ 0\) interpolation functions to the definitions of certain curved finite elements.

Chapter 7 marks the beginning of the second part of the book, the one which deals with dynamic analysis problems. This chapter dwells on parabolic, hyperbolic and elliptic eigenvalue problems and their applications in elastostatics, heat conduction, viscous fluid flow, the study of variations, and stability analysis. In chapter 8 the author puts forth algorithms for solving first-order nonstationary problems - parabolic problems. The Euler and Crank-Nicolson methods are presented, and the convergence and stability of the solutions are improved. Algorithms are presented for solving this category of problems by means of element-by-element methods.

Chapter 9 includes algorithms for solving second-order nonstationary problems - hyperbolic and parabolic-hyperbolic problems. The author describes and compares the Houbolt, Newmark, Park, Wilson and predictor- corrector approaches. Chapter 10 discusses numerical procedures for solving eigenvector and eigenvalue problems: the Jacobi method, subspace iteration, the Lanczos algorithm, the Rayleigh-Ritz and Irons-Guyan reduction techniques, and static condensation. The author also describes a package of subroutines written in FORTRAN 77 and based on the Lanczos method.

Chapter 11 presents the computer program DLEARN, which is written in FORTRAN 77 and implements the theoretical and numerical concepts introduced in chapters 3, 5, 7, 9, and 10. The program is intended to deal with linear static and dynamic analysis problems and involves two types of finite elements - the four-node quadrilateral, elastic continuum element and the three-dimensional, elastic truss element. The structures of the program is modular and multipurpose, its open-ended architecture enabling the user to extend and/or alter the program.

The book is undoubtedly original as far as both overall presentation and the content of certain chapters (particularly chapter 4, but also chapters 5, 6, 10, and 11) are concerned. It can safely be concluded that this book goes beyond its avowed purpose - that of providing introductory material to the students who want to study the FEM - and is a valuable addition to the list of reference works covering this field, since the author not only presents the method in question, but also comes up with new developments.

Reviewer: V.Goincu

##### MSC:

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

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

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

65K10 | Numerical optimization and variational techniques |

76M99 | Basic methods in fluid mechanics |