Scaled Boundary Finite Element Method

Abstract

5R3. Scaled Boundary Finite Element Method. - JP Wolf (Swiss Fed Inst of Tech, Lausanne, Switzerland). Wiley, W Sussex, UK. 2003. 361 pp. ISBN 0-471-48682-5. $130.00.Reviewed by Long-Yuan Li (Dept of Civil Eng, Aston Univ, Aston Triangle, Birmingham, B4 7ET, UK).This book describes a fundamental solution-less boundary element method, based on finite elements. The method combines the advantages of both the finite and boundary element methods as the finite element discretization in the method is restricted to the circumferential direction while in the radial direction it uses a scaling procedure to obtain an analytical solution. The method can be used to analyze any bounded and unbounded media governed by linear elliptic, parabolic, and hyperbolic partial differential equations. The book is based on the research and development performed recently by the author and his colleagues. It is a unique research book that presents the development of new numerical procedures that can overcome some difficulties that appear when using the finite or boundary element method. The book contains 26 chapters, 4 appendices, high quality figures, and a good subject index. References are provided in the alpha beta order, which are listed at the end of the chapters. The first two chapters provide a brief introduction of numerical procedures and features of the finite element method, boundary element method, and scaled boundary finite element method. More details of the concepts of the scaled boundary finite element method and its applications in model problems and two- and three-dimensional elastodynamic, static’s and diffusion problems are presented in Part I and II. Part I addresses the model problem, which contains 12 chapters (Chapters 3-14). Chapter 3 addresses the concepts of scaled boundary transformation of geometry and similarity. Chapter 4 gives the definition of a model problem. Two derivations of scaled boundary finite element equations are presented. In Chapter 5 the weighted-residual technique is used, and the other in Chapter 6 uses the similarity and finite element assemblage. Chapter 7 discusses the analytical solution of the scalar scaled boundary finite element equations. Chapters 8-12 discuss the solution procedures of the scaled boundary finite element equations in displacement and in dynamic stiffness for bounded and unbounded media. In Chapter 13, implementation issues are discussed, which also apply to the general matrix equations. Chapter 14 gives the conclusions related to the model problem. At the end of Part I, four short appendices are provided, leading to deeper insight into certain aspects of the model problem, and providing a link to the generalization of two- and three-dimensional static’s, elastodynamics and diffusion in Part II. Appendix A deals with solid modeling, Appendix B discusses the analysis in the frequency domain, Appendix C establishes the equations of motion of a dynamic unbounded medium-structure interaction problem using the properties calculated in the Model Problem, and Appendix D describes the early historical development leading up to the scaled boundary finite element method. Part II has 12 chapters (Chapters 15 to 26), which develops all aspects of the current state of the art of the scaled boundary finite element method. Following the derivation of the fundamental equations based on the scaled boundary transformation (described in Chapter 15), the solution procedures for static’s and dynamics in the frequency and time domains, both numerically and analytically, for bounded and unbounded media are developed in Chapters 16-22, respectively. Two- and three-dimensional examples in elastodynamics and diffusion for bounded and unbounded media are discussed in Chapters 23 and 24. Based on the stress recovery technique error estimation and adaptivity are discussed in Chapter 25. Chapter 26 contains concluding remarks and addresses restrictive properties of the novel method and suggestions for future research. In summary, Scaled Boundary Finite Element Method, is a self-contained, well-presented advanced textbook. It is suitable for research students and for the personal bookshelves of research investigators working in the field of computational engineering sciences. It can also be a useful reference in libraries.