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Finite element methods for computational fluid dynamics : a practical guide
Kuzmin D., Hämäläinen J., SIAM, Philadelphia, PA, 2014. 321 pp. Type: Book (978-1-611973-60-0)
Date Reviewed: Aug 28 2015

Kuzmin and Hämäläinen’s book provides a general overview of finite element methods (FEMs) when applied to computational fluid dynamics (CFD), giving a gentle introduction to these fields, a survey of recent advanced works in the literature, and a practical guide to Elmer, a FEM toolbox developed by the Finnish CSC-IT Centre for Science. CFD is the scientific discipline that deals with the computer simulation of gas and liquid behaviors, as in the distribution of the fuel-air mixture in a jet engine, the air flow patterns around a car, or the transport of oil inside a pipeline; it is at the heart of contemporary industrial design and engineering. Those physical phenomena are mathematically described by partial differential equations (PDEs) ∂u/∂t + f(u) = s, where the fields u(x,t) represent speed, temperature, or pressure values in a given system. Yet before these equations can be solved on today’s digital computers, such fields must be discretized. Finite difference schemes approximate derivatives by ratios of small differences over discrete Cartesian grids. FEMs use a more abstract approach, where the fields u(x,t) are seen as solutions of a constraint problem, namely that of zeroing the scalar products of the residuals r(u) = ∂u/∂t + f(u) - s of the PDEs of interest with specific test functions w(x). The discretization process then proceeds by decomposing each field u(x,t) into a weighted sum of elementary basis functions (typically linear or bilinear and belonging to the same functional space as the test functions), all defined over only very small, non-overlapping domains, ending up covering the simulated space. This way, the decomposition of the physical parameters can be performed over grids of complex topologies, tuned to the particular shape of the objects under study. Nullifying the mentioned scalar products yields the best values for the weights used to represent the pressure, temperature, or speed fields. This process can be shown to be equivalent to solving an algebraic system of the form M du/dt +Ku = r, where u and r are now vectors of values of the field at each node of the unstructured grid and the matrices M and K are defined using only scalar products of basis functions. After applying any of the existing time discretization techniques, for example, a finite difference scheme, one ends up with a fully discrete problem that amounts to solving, for each time stamp, a linear system Aut = b, where A is usually a sparse matrix and b depends on both the initial conditions and the value of ut-1 at the previous time step.

Even though the overall emphasis is on practical concerns, this nine-chapter monograph also deals with the precise mathematical manipulations that put the overall FEM framework on solid ground. After a fast-paced preface that includes a list of key mathematical definitions and notions, the introductory chapter puts the use of FEMs for computationally solving the transport equation of fluid mechanics with convection and diffusion in perspective with over frameworks such as the finite difference and finite volume methods. Key indicators of reliable simulations such as the maximum principles (that is, ensuring that the approximate solutions remain within bounds linked to the initial conditions) are also introduced. The bread and butter of CFD is the Navier-Stokes equations, which encode the conservation of fluid mass, momentum, and energy; they are presented in chapter 3. The basics of FEM are illustrated then with the traditional 1D heat equation, for which the generic ten-step Galerkin algorithm that structures FEM is presented in detail. Chapter 5 logically extends this general framework to the 2D case, and ends with a tutorial on the use of Elmer in such a setting on real examples. If the Galerkin method yields good results in purely diffusive situations, the introduction of convection induces numerical artifacts (oscillations, instabilities) that must be addressed to recover usable simulations; the next two chapters describe in detail how such issues are handled by variants of the Galerkin method (Petrov-Galerkin and Taylor-Galerkin in their 1D and nD versions, in which artificial diffusion is introduced to stabilize these spurious effects. The final chapter provides a rapid glance at more advanced techniques dedicated to incompressible flow problems; in fact, this chapter is a genuine entry gate to current research in FEMs for CFD, with its numerous references to quite recent academic literature. Further reading suggestions and a voluminous bibliography end the monograph.

This book packs a whole lot of information into about 300 pages. A large range of readers will appreciate a clarity that does not shy away from formal developments, as well as the use of computational techniques that correspond to the state of the art in numerical methods for CFD. It can also be used as a first introduction to FEM software tools via a few practical examples; those can be solved using the Elmer open-source package.

Reviewer:  P. Jouvelot Review #: CR143729 (1511-0938)
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