Hybrid And Mixed Finite Element Methods

Research on non-standard finite element methods is evolving rapidly and in this text Brezzi and Fortin give a general framework in which the development is taking place. The presentation is built around a few classic examples: Dirichlet's problem, Stokes problem, Linear elasticity. The authors provide with this publication an analysis of the methods in order to understand their properties as thoroughly as possible.

Non-standard finite element methods, in particular mixed methods, are central to many applications. In this text the authors, Boffi, Brezzi and Fortin present a general framework, starting with a finite dimensional presentation, then moving on to formulation in Hilbert spaces and finally considering approximations, including stabilized methods and eigenvalue problems. This book also provides an introduction to standard finite element approximations, followed by the construction of elements for the approximation of mixed formulations in H(div) and H(curl). The general theory is applied to some classical examples: Dirichlet's problem, Stokes' problem, plate problems, elasticity and electromagnetism.

While the theory and application of finite elements methods can be extended to incompatible, hybrid, and mixed element methods, important issues, such as determining the reliability of the solution of incompatible multivariable elements, along with a common perception of impracticality, have hindered the widespread implementation of these methods.

Mixed finite element methods are a tool to solve complex engineering problems of different nature. This subject is treated in the volume from the engineering and the mathematical point. Different applications are considered which depict the value of mixed formulations in engineering on one side. On the other side the mathematical background is provided including proofs of convergence and stability of these methods and adequate solvers for mixed problems are discussed. This broad spectrum yields an indepth treatment of mixed methods from different perspectives.

In this book, based on 16 years of work on the finite element method, the author presents the essence of a new, direct approach to the FEM. The work is focused on the mixed method and shows how reliable results may be obtained with fewer equations than usual. The basic principles, the fundamentals and the essence of the FEM are presented, then the method is applied to the analysis of one, two, and three-dimensional problems. It is shown that mixed elements offer superior accuracy compared with stiffness elements. Finally, some new achievements and perspectives for further development are presented. The book is intended for undergraduate and graduate students, mathematicians, research engineers and practicing engineers. To understand the book, a familiarity with classical mechanics is sufficient.

This book has one single purpose: to present the development of the partial hybrid finite element method for the stress analysis of laminated composite structures. The reason for this presentation is because the authors believe that partial hybrid finite element method is more efficient that the displacement based finite element method for the stress analysis oflaminated composites. In fact, the examples in chapter 5 of this book show that the partial hybrid finite element method is about 5 times more efficient than the displacement based finite element method. Since there is a great need for accurate and efficient calculation of interlaminar stresses for the design using composites, the partial hybrid finite method does provide one possible solution. Hybrid finite method has been in existence since 1964 and a significant amount of work has been done on the topic. However, the authors are not aware of any systematic piece of literature that gives a detailed presentation of the method. Chapters of the displacement finite element method and the evolution 1 and 2 present a sununary of the hybrid finite element method. Hopefully, these two chapters can provide the readers with an appreciation for the difference between the displacement finite element method and the hybrid finite element. It also should prepare the readers for the introduction of partial hybrid finite element method presented in chapter 3.