3R2. Mesh Free Methods: Moving Beyond the Finite Element Method. - GR Liu (Center for Adv Computations in Eng Sci, Natl Univ of Singapore). CRC Press LLC, Boca Raton FL. 2003. 691 pp. ISBN 0-8493-1238-8. \$149.95.

Reviewed by D Karamanlidis (Dept of Civil Eng, Univ of Rhode Island, Bliss Hall, Kingston, RI 02881).

As anyone who has ever used finite elements over a prolonged period of time knows, the method does have its share of shortcomings. Hence, in the early 1980s commercial software started popping up, which was based on boundary elements—elements, that is, defined only over the boundary of the problem under consideration. Mesh free methods attempt to “move beyond the Finite and Boundary Element Methods” in as much as they abandon the element (subregion) concept altogether and use instead a set of nodal points to discretize the domain. Now, according to the author’s preface, “…the book provides systematic steps that lead the reader to understand mesh free methods, how they work, how to use and develop a mesh free method, as well as the problems, associated with the element free methods…” The author further states that the book is intended for “…senior university students, graduate students, researchers, and professionals in engineering and science. Mechanical engineers and practitioners and structural engineers and practitioners will also find the book useful.”

The book is quite voluminous and covers a whole range of engineering mechanics problems both for solids and fluids. The author certainly did a commendable job in preparing a single source that presents not only his own research work, but also the accomplishments by others working in this relatively new area of computational mechanics. Of the 16 chapters of which the book is comprised, the first two serve as an introduction, the third gives a short presentation of the equations of linear elasticity, the following 12 outline, in great detail, the various possible approaches in developing mesh free methods. Along with the theoretical background, numerous examples are presented that highlight the performance of mesh-free versus mesh-based methods. In the last chapter, a software for 2D analyses based on mesh free techniques is presented.

Overall, the book is well-written, the figures and illustrations are very good, the list of references is quite comprehensive, and the subject index is useful. The overall impression of the book is somewhat diminished by a large number of inaccuracies, which were not detected in the proofs. Some examples follow:

On page 148, “sensor” should read “tensor.”

On page 148, “displacement comment” should read “displacement component.”

On page 161, Eq. (6.51) is attributed to Timoshenko and Goodier. However, the equation found on page 44 of that reference is quite different. The same applies to the formulas presented on page 162 of the present book. For example, Eq. (6.54) states that the normal stress on the cross section of the beam is
$σx=−Ps˙L−x/I$
Obviously, the variable y is missing.

On page 219, “numerical investigation” should read “numerical integration.”

On page 240, $0⩽t⩽Δt$ should read $0⩽τ⩽Δt.$

On page 404, “bulking” should read “buckling.”

Also, some of the statements made in the book seem confusing if not wrong. For example,

“…For centuries, people have been using the finite difference method…” While, older than FEM, FDM has certainly not been used for centuries.

“…Hamilton’s principle is…based on the energy principle.” Hamilton’s principle is valid for both conservative and non-conservative systems for which no energy principle exists.

“Natural and essential boundaries.” They are more commonly referred to as boundaries on which natural and respectively essential conditions are prescribed.

The term of “stability” in conjunction with a mesh-free local discretization does not appear to have been defined in the book.

“The Lagrange multipliers…can be viewed physically as smart forces that can force $u−u¯=0.$ If the trial function can be so chosen that $u−u¯=0,$ the smart force will be zero…” For one “smart force” is a bit unusual terminology. As one sees from the pertinent Euler equations, the Lagrange multipliers are the support reactions. More importantly though, satisfying the geometric boundary conditions does not make the Lagrange multipliers themselves but the work produced by them zero.

The heading of Section 6.4.1, “Basic Equations for Nonlinear Mechanics Problems,” is misleading insofar that the following equations are not correct for a general nonlinear problem involving material and geometric nonlinearities.

The above notwithstanding, Mesh Free Methods: Moving Beyond the Finite Element Method is recommended for purchase to libraries and researchers who have a solid background in discretization methods and are interested in learning about mesh free methods.