Introduction to finite element methods in computational fluid dynamics

Abstract

The finite element method (FEM) is a numerical technique for solving partial differentialequations (PDE's). Its first essential characteristic is that the continuum field,or domain, is subdivided into cells, called elements, which form a grid. The elements(in 2D) have a triangular or a quadrilateral form and can be rectilinear or curved. Thegrid itself need not be structured.With unstructured grids and curved cells, complexgeometries can be handled with ease. This important advantage of the method is notshared by the finite difference method (FDM) which needs a structured grid, which,however, can be curved. The finite volume method (FVM), on the other hand, hasthe same geometric flexibility as the FEM.The second essential characteristic of the FEM is that the solution of the discreteproblem is assumed a priori to have a prescribed form. The solution has to belongto a function space, which is built by varying function values in a given way, forinstance linearly or quadratically, between values in nodal points. The nodal points,or nodes, are typical points of the elements such as vertices, mid-side points, midelementpoints, etc. Due to this choice, the representation of the solution is stronglylinked to the geometric representation of the domain. This link is, for instance, notas strong in the FVM.The third essential characteristic is that a FEM does not look for the solution ofthe PDE itself, but looks for a solution of an integral form of the PDE. The mostgeneral integral form is obtained from a weighted residual formulation. By this formulationthe method acquires the ability to naturally incorporate differential typeboundary conditions and allows easily the construction of higher order accuratemethods. The ease in obtaining higher order accuracy and the ease of implementationof boundary conditions form a second important advantage of the FEM. Withrespect to accuracy, the FEM is superior to the FVM, where higher order accurateformulations are quite complicated.The combination of the representation of the solution in a given function space,with the integral formulation treating rigorously the boundary conditions, gives tothe method an extremely strong and rigorous mathematical foundation.A final essential characteristic of the FEM is the modular way in which the discretizationis obtained. The discrete equations are constructed from contributions onthe element level which afterwards are assembled.Historically, the finite element method originates from the field of structural mechanics.This has some remnants in the terminology. In structural mechanics, thepartial differential formulation of a problem can be replaced by an equivalent variationalformulation, i.e. the minimization of an energy integral over the domain.The variational formulation is a natural integral formulation for the FEM. In fluiddynamics, in general, a variational formulation is not possible. This makes it lessobvious how to formulate a finite element method. The history of computationalfluid dynamics (CFD) shows that every essential break-through has first been madein the context of the finite difference method or the finite volume method and thatit always has taken considerable time, often much more than a decade, to incorporatethe same idea into the finite element method. The history of CFD, on the otherhand, also shows that, once a suitable FEM-formulation has been found, the FEM isalmost exclusively used. This is due to the advantages with respect to the treatmentof complex geometries and obtaining higher order accuracy.The development of the finite element method in fluid dynamics is at present stillfar from ended. For the simplest problems such as potential flows, both compressibleand incompressible, and incompressible Navier-Stokes flows at low Reynoldsnumbers, the finite element method is more or less full-grown, although new evolutions,certainly for Navier-Stokes problems, are still continuing. More complexproblems like compressible flows governed by Euler- or Navier-Stokes equations orincompressible viscous flows at high Reynolds numbers still form an area of activeresearch.In this introductory text, the option is taken to explain the basic ingredients ofthe finite element method on a very simple, purely mathematical, problem and togive fluid dynamics illustrations in detail only for simple problems. For more complexproblems, only a basic description is given with reference to further literature.Also in the explanation of the method, all mathematical aspects are systematicallyavoided. For the mathematical aspects, reference is made to further literature. Thismakes the text accessible for a reader with almost no knowledge of functional analysisand numerical analysis. For the fluid dynamics illustrations, the option has beentaken to use only simple techniques, so that the detailed examples can be reproducedby the reader not really familiar with general computational fluid dynamics or evengeneral fluid dynamics. This text therefore is to be seen as the absolute minimumintroduction to the subject. The text is in no way complete and the author deliberatelyhas taken the risk to be seen as naive by a more informed reader. A referencelist is given for a deeper introduction. A reader beginning with computational fluiddynamics should be aware that a complete study of the finite element method maytake considerable time and may necessitate, depending on background, a considerableeffort. The method is much less intuitive than the finite difference method andthe finite volume method and requires a more fundamental attitude for mathematicalformulations. This introductory text therefore is also meant to create some enthusiasmfor the method by showing its power with simple examples and to justify in thisway the need for further study. It is the conviction of the author that a practitioner ofCFD, even if it is not his or her intention to use the FEM, should have at least a basicknowledge of the method. This is in particular useful with respect to the treatmentof boundary conditions. Also one should consider that the impact of the FEM inCFD is already extremely important and that it probably will grow in the future. © Springer-Verlag Berlin Heidelberg 2009.