Book contents
- Frontmatter
- Contents
- Preface
- PART ONE PRELIMINARIES
- PART TWO FINITE DIFFERENCE METHODS
- PART THREE FINITE ELEMENT METHODS
- 8 Introduction to Finite Element Methods
- 9 Finite Element Interpolation Functions
- 10 Linear Problems
- 11 Nonlinear Problems/Convection-Dominated Flows
- 12 Incompressible Viscous Flows via Finite Element Methods
- 13 Compressible Flows via Finite Element Methods
- 14 Miscellaneous Weighted Residual Methods
- 15 Finite Volume Methods via Finite Element Methods
- 16 Relationships between Finite Differences and Finite Elements and Other Methods
- PART FOUR FOUR. AUTOMATIC GRID GENERATION, ADAPTIVE METHODS, AND COMPUTING TECHNIQUES
- PART FIVE APPLICATIONS
- APPENDIXES
- Index
8 - Introduction to Finite Element Methods
Published online by Cambridge University Press: 15 January 2010
- Frontmatter
- Contents
- Preface
- PART ONE PRELIMINARIES
- PART TWO FINITE DIFFERENCE METHODS
- PART THREE FINITE ELEMENT METHODS
- 8 Introduction to Finite Element Methods
- 9 Finite Element Interpolation Functions
- 10 Linear Problems
- 11 Nonlinear Problems/Convection-Dominated Flows
- 12 Incompressible Viscous Flows via Finite Element Methods
- 13 Compressible Flows via Finite Element Methods
- 14 Miscellaneous Weighted Residual Methods
- 15 Finite Volume Methods via Finite Element Methods
- 16 Relationships between Finite Differences and Finite Elements and Other Methods
- PART FOUR FOUR. AUTOMATIC GRID GENERATION, ADAPTIVE METHODS, AND COMPUTING TECHNIQUES
- PART FIVE APPLICATIONS
- APPENDIXES
- Index
Summary
GENERAL
The finite element theory as applied to one-dimensional problems was discussed in Part One, Preliminaries. In general, finite element methods (FEM) are versatile in applications to multidimensional complex irregular geometries. Initial applications of FEM began with structural analysis in the late 1950s and primarily were based on variational principles. Duringthe early days of the development of FEM, applications were made for simple flow problems, beginning with Zienkiewicz and Cheung [1965], followed by Oden and Wellford [1972], Chung[1978], and Baker [1983], amongothers. Signifi-cant contributions in CFD began with the streamline upwind Petrov-Galerkin (SUPG) methods [Heinrich, Huyakorn, Zienkiewicz, and Mitchell, 1977; Hughes and Brooks, 1982; Hughes, Mallet, and Mizukami, 1986] or streamline diffusion methods (SDM) [Johnson, 1987], Taylor-Galerkin methods (TGM) [Donea, 1984; Löhner, Morgan, and Zienkiewicz, 1985], and hp adaptive methods [Oden and Demkowicz, 1991], among many other related works.
New approaches and various alternative methodologies are preponderant in the literature. Efforts are made in this book to simplify and unify some of the terminologies. For example, the original approaches of SUPG or SDM for convection-dominated flows have grown into GLS (Galerkin/least squares) when some changes in the formulation are introduced. It is suggested that all methods related to numerical diffusion test functions be called the generalized Petrov-Galerkin (GPG) methods. Hughes and his co-workers have contributed significantly in the past two decades to the GPG methodologies associated with the problems of convection-dominated flows and shock discontinuities.
Another example is the algorithm arising from the Taylor series expansion such as TGM. Zienkiewicz and his co-workers [Zienkiewicz and Codina, 1995] have applied for the past decade the concept of characteristic Galerkin methods (CGM) which produce results similar to TGM in dealing with convection-dominated problems for both compressible and incompressible flows.
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- Computational Fluid Dynamics , pp. 243 - 261Publisher: Cambridge University PressPrint publication year: 2002