Finite elements: theory, fast solvers, and applications in solid mechanics [Book and Web reviews]

81 T his fascinating book deals with the mathematical aspects of the finite element method. Engineers developed the FEM in the early 1960s to solve structural mechanics problems— it was based initially on energy theorems applicable to elastic structures. Soon, engineers realized that they could extend the initial methodology to cover the numerical solution of general partial differential equations. This method attracted the attention of mathematicians , who studied its mathematical aspects and developed a rigorous basis for it. Even though many scientific papers deal with the method's mathematical aspects , a vast majority of books deal with the FEM approach from an " engineering " point of view and focus on its practical aspects. Braess's book presents the FEM as a general method for the solution of partial-differential equations, discusses in detail its mathematical foundations, and contributes successfully to the relatively limited bibliography on the subject. Although complete references to original works appear throughout the book, the section that discusses C1 triangular elements (pp. 65–67) describes the Argyris triangle and the Hsieh-Clough-Tocher element but gives no further information that can guide you to the original sources. I hope that any new editions of the book will correct this. A common " variational crime " committed by finite element solutions is the numerical evaluation of integrals using quadrature formulas involving point-evaluation functionals that are not defined for the functions involved. This leads to nonconforming elements, which a separate chapter covers in detail. In finite element solutions, it is also common to enforce constraints, such as material incompressibility, only in a weak sense. This takes into account a finite number of the infinite constraints of the problem and leads to saddle-point problems and " mixed " finite element solutions. Braess discusses the well-known Ladyshenskaya-Babuska-Brezzi condition, which is important for the proper treatment of mixed formulations , where common sense usually leads to the development of elements that exhibit unstable behavior. The book closes with a chapter on structural-mechanics applications that starts with a nice, concise review of the theory of elasticity. This is very helpful , and makes this chapter attractive to people who are not experts in solid mechanics. Certain problems in mechanics involve a small parameter, say t, which can greatly influence the numerical solution's quality. For example, the thickness of a membrane is much smaller than the other dimensions, or the change in density of a metal during …