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IsoGeometric analysis : a new paradigm in the numerical approximation of PDEs
Buffa A., Sangalli G., Springer International Publishing, New York, NY, 2016. 193 pp. Type: Book (978-3-319423-08-1)
Date Reviewed: Feb 22 2017

For their daily work, computational engineers in many technical disciplines rely heavily on the use of finite element software systems. However, certain features of the traditional mathematical approaches on which these systems are based impose limitations on today’s use cases that can be highly undesirable. The isogeometric analysis concept is a relatively novel set of ideas that can help overcome many of these limitations. This volume presents a nicely readable introduction to these lines of thought.

Specifically, the conventional approach behind a finite element algorithm is based on a geometric description of the object whose behavior is to be simulated. This geometry is first discretized into a mesh of polygons or polyhedra. Once these so-called elements have been determined, it is rather simple and straightforward, from a mathematical point of view, to set up the required equations on each of them. Unfortunately, the construction of the elements is by no means easy, in particular because the underlying geometries tend to become more and more involved. Thus, the constraints on the elements that need to be satisfied in order to obtain sufficiently accurate simulation results from the algorithms are very difficult to fulfill. The isogeometric approach attempts to circumvent these difficulties by setting up the finite element equations directly on the given geometry, thus avoiding the discretization step. The successful implementation of such a procedure consists of a number of individual steps; each of these steps is addressed in one chapter of the book.

The first chapter contains an introduction to computer-aided geometric design (CAGD). The authors present the basic concepts for representing curves and surfaces, such as Bernstein polynomials and B-splines. These concepts will later prove to be fundamental for constructing the finite element equations. The chapter is written in a tutorial style and describes all the important points without requiring significant prerequisites on the reader’s side.

Chapter 2 is by a large amount the shortest chapter of the book. It describes how the given geometric objects can be decomposed into smaller subobjects suitable for later numerical work without following the traditional path of leaving the given geometry. Particular attention is paid to the question of smooth transitions between neighboring subobjects. In this sense, chapter 2 builds the bridge between the geometrical concepts described in chapter 1 and the numerical and analytical aspects of solving the given model equations that are discussed in chapters 3 and 4.

The main topic of chapter 3 is the mathematical analysis of the problem. Recalling some of the facts already mentioned in chapter 1, the authors explain how to set up the differential equations of the model in question on the specific domain and provide an analytical derivation of the numerical approximation method. Implementation-related aspects and error estimates are also briefly discussed.

Finally, chapter 4 presents the application of an isogeometric analysis algorithm on a real-world problem, thus demonstrating the capabilities of the approach.

Unfortunately the book does not have a subject index. This seems to be its only major drawback. On the positive side, all chapters are illustrated with many high-quality figures, most of them in color. This greatly helps the reader understand the underlying concepts.

Although the book was published in 2016, it is the outcome of a summer school held in 2012. Some of the chapters present the state of the art of 2012 and so do not address the newest developments, whereas others appear to be more up to date. However, since the book is of an introductory nature, I do not consider this to be a major disadvantage. I fully recommend it as good material for traditionally educated finite element experts who want to extend their knowledge to this new area, and also for finite element novices who want to get into the field.

Reviewer:  Kai Diethelm Review #: CR145075 (1705-0247)
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