The complex variable boundary element method in engineering analysis.

*(English)*Zbl 0609.65078
New York etc.: Springer-Verlag. VIII, 389 p.; DM 88.00 (1987).

This book introduces an alternative boundary element like formulation for the solution of two-dimensional potential theory problems. The technique is called the Complex Variable Boundary Element Method (CVBEM) and is based in a generalization of the Cauchy integral formula, which is solved in a boundary element fashion procedure (i.e. only boundary approximations over elements are included into the numerical solution). The procedure stems from the so-called Analytic Function Method, which was introduced by the end of the last decade, and as opposed to the standard BEM formulations for the same problems, requires the use of complex variable theory for its formulation.

Chapter one introduces the basic principles and the governing equations for many physical problems which fall into the field of potential theory. Chapter two reviews the complex variable theory and presents the mathematical background required for the formulation of the method. Chapter three is devoted to the actual formulation of the technique and presents a detailed description of each of the main mathematical aspects associated with the method. The numerical features of the procedure are taken further in chapter four where computer algorithms for the implementation are presented and discussed in detail.

Chapter five introduces useful means of reducing the boundary errors associated with the numerical approximation and chapter six discusses the error analysis capabilities offered by the CVBEM. Here, the introduction of an approximate boundary which corresponds to the exact boundary associated with the numerical solution is found to be a useful tool for controlling the approximation of the numerical solution.

Chapter seven is devoted to practical applications and more sophisticated modelling techniques, such as the possibility of including sources or sinks and regional inhomogeneities (subregions) in the solution procedure. A number of applications and combinations of CVBEM with other solution methods is presented in chapter eight.

Throughout the book a wealth of FORTRAN computer programs is presented to illustrate the different algorithms introduced. A missing point, however, is some information concerning the performance of the method (i.e. accuracy and computer efficiency) in comparison with other numerical techniques such as the standard direct BEM.

The publication is sufficiently thorough and carefully written, providing excellent means of getting acquainted with this new alternative boundary element formulation which is restricted to two-dimensional applications.

Chapter one introduces the basic principles and the governing equations for many physical problems which fall into the field of potential theory. Chapter two reviews the complex variable theory and presents the mathematical background required for the formulation of the method. Chapter three is devoted to the actual formulation of the technique and presents a detailed description of each of the main mathematical aspects associated with the method. The numerical features of the procedure are taken further in chapter four where computer algorithms for the implementation are presented and discussed in detail.

Chapter five introduces useful means of reducing the boundary errors associated with the numerical approximation and chapter six discusses the error analysis capabilities offered by the CVBEM. Here, the introduction of an approximate boundary which corresponds to the exact boundary associated with the numerical solution is found to be a useful tool for controlling the approximation of the numerical solution.

Chapter seven is devoted to practical applications and more sophisticated modelling techniques, such as the possibility of including sources or sinks and regional inhomogeneities (subregions) in the solution procedure. A number of applications and combinations of CVBEM with other solution methods is presented in chapter eight.

Throughout the book a wealth of FORTRAN computer programs is presented to illustrate the different algorithms introduced. A missing point, however, is some information concerning the performance of the method (i.e. accuracy and computer efficiency) in comparison with other numerical techniques such as the standard direct BEM.

The publication is sufficiently thorough and carefully written, providing excellent means of getting acquainted with this new alternative boundary element formulation which is restricted to two-dimensional applications.

Reviewer: J.C.F.Telles

##### MSC:

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65R20 | Numerical methods for integral equations |

65E05 | General theory of numerical methods in complex analysis (potential theory, etc.) |

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

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

35C15 | Integral representations of solutions to PDEs |

31C20 | Discrete potential theory |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35J40 | Boundary value problems for higher-order elliptic equations |