The Finite Element Method, Volumes 1, 2 and 3, Fluid Dynamics, 5^th Edition
by O. C. Zienkiewicz and R. L. Taylor 

The fifth edition of The Finite Element Method, by O. C. Zienkiewicz and R. L. Taylor has grown from a single volume to a three-volume set of 1440 pages. Every previous edition of The Finite Element Method has rightly been regarded as a classic, and this edition is no different.  Without question, this set of books is the most comprehensive text on the finite element method available, and will likely remain so for the foreseeable future.

Unlike many books on the finite element method, this set is written with the assumption that the reader hasn’t had any previous exposure to the finite element method or mathematical training beyond basic calculus and linear algebra. While the text is advertised as being for graduate students and researchers, it is accessible to motivated junior and senior undergraduates. The mathematical notation is always very clear, as is the writing.

The one-volume, third edition of The Finite Element Method was the textbook for the first finite element course I took at The University of Michigan. Starting with the modeling of a simple discrete system, it introduced the essential ideas of the finite element method in a few hundred pages. At over 700 pages, it was far too long to cover in a single semester, or even an academic year. I viewed it then, and I still do today, as a comprehensive text on all aspects of linear finite element theory. The first volume of the fifth edition fulfills the same role as the earlier, single volume edition.

The second volume on solid mechanics covers both linear and nonlinear problems in solid mechanics. Professor Taylor’s contributions to shell elements are well known, and won’t be repeated here. Needless to say, this is probably the book for linear shell elements. The coverage of nonlinear problems is more extensive than in the two-volume, sixth edition.

The third volume covers fluid dynamics. As a text in finite element methods for fluid dynamics, it is very good.  Individuals whose primary interest is computational fluid dynamics will want to purchase additional texts since this volume does not delve deeply into the finite difference and finite volume methods that are at the heart of many commercial fluids programs.

For students looking for an introduction to the finite element method, they couldn’t go wrong choosing Volume 1 of The Finite Element Method. Serious students and researchers will probably want to purchase the entire set and save a few dollars over purchasing the volumes separately.

The Finite Element Method, Volume 1, The Basis, 5^th Edition
 by O. C. Zienkiewicz and R. L. Taylor -
Available for purchase at Amazon  736 pages, $95.00

Contents:

• Some preliminaries: The standard discrete system

• A direct approach to problems in elasticity

• Generalization of the finite element concepts – Galerkin-weighted residual and variational approaches

• Plane stress and plane strain

• Axisymmetric stress analysis

• Three-dimensional stress analysis

• Steady-state field problems – heat conduction, electric and magnetic potential, fluid flow etc

• 'Standard' and 'hierarchical' element shape functions: some general familiarities of C0 continuity

• Mapped elements and numerical integration - 'infinite' and 'singularity' elements

• The patch test, reduced integration and non-conforming elements

• Mixed formulation and constraints – complete field methods

• Incompressible problems, mixed methods and other procedures of solution

• Mixed formulation and constraints – incomplete (hybrid)field methods, boundary Trefftz methods

• Errors, recovery processes and error estimates

• Adaptive finite element refinement

• The time dimension - semi-discretization of field and dynamic problems and analytical solution procedures

• The time dimension - discrete approximation in time

• Coupled systems Computer procedures for finite element analysis

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The Finite Element Method, Volume 2, Solid Mechanics, 5th Edition
 by O. C. Zienkiewicz and R. L. Taylor 
Available for purchase at Amazon 384 pages, $85.00

Contents:

• General Problems in Solid Mechanics and Non-Linearity.

• Solution of Non-Linear Algebraic Equations.

• Inelastic and Non-Linear Materials.

• Plate Bending Approximation; Thin (Kirchhoff) Plates and C1 Continuity Requirements.

• 'Thick' Reissner-Mindlin Plates - Irreducible and Mixed Formulations.

• Shells as an Assembly of Flat Elements.

• Axisymmetric Shells.

• Shells as a Special Case of Three-Dimensional Analysis - Reissner-Mindlin Assumptions.

• Semi-Analytical Finite Element Processes - Use of Orthogonal Functions and 'Finite Strip' Methods.

• Geometrically Non-Linear Problems - Finite Deformation.

• Non-Linear Structural Problems - Large Displacement and Instability.

• Pseudo-Rigid and Rigid-Flexible Bodies.

• Computer Procedures for Finite Element Analysis.

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The Finite Element Method, Volume 3, Fluid Dynamics, 5^th Edition
 by O. C. Zienkiewicz and R. L. Taylor 
Available for purchase at Amazon, 320 pages, $85.00

Contents:

• Introduction and the Equations of Fluid Dynamics.

• Convection Dominated Problems - Finite Element Approximations to the Convection-Diffusion Equation.

• A General Algorithm for Compressible and Incompressible Flows - The Characteristic-Based Split (CBS) Algorithm.

• Incompressible Laminar Flow - Newtonian and Non-Newtonian Fluids.

• Free Surfaces, Buoyancy and Turbulent Incompressible Flows.

• Compressible High-Speed Gas Flow.

• Shallow-Water Problems.

• Waves.

• Computer Implementation of the CBS Algorithm.

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The Finite Element Method, Volumes 1, 2 and 3, Fluid Dynamics, 5^th Edition
by O. C. Zienkiewicz and R. L. Taylor 
Available for purchase at Amazon $225.00