Book contents
- Frontmatter
- Brief Contents
- Contents
- Preface
- 1 The Finite Element Method: Introductory Remarks
- 2 Some Methods for Solving Continuum Problems
- 3 Variational Approach
- 4 Requirements for the Interpolation Functions
- 5 Heat Transfer Applications
- 6 One-Dimensional Steady-State Problems
- 7 The Two-Dimensional Heat-Conduction Problem
- 8 Three-Dimensional Heat-Conduction Applications with Convection and Internal Heat Absorption
- 9 One-Dimensional Transient Problems
- 10 Fluid Mechanics Finite Element Applications
- 11 Use of Nodeless Degrees of Freedom
- 12 Finite Element Analysis in Curvilinear Coordinate
- 13 Finite Element Modeling of Flow in Annular Axisymmetric Passages
- 14 Extracting the Finite Element Domain from a Larger Flow System
- 15 Finite Element Application to Unsteady Flow Problems
- 16 Finite Element-Based Perturbation Approach to Unsteady Flow Problems
- Appendix A Natural Coordinates for Three-Dimensional Surface Elements
- Appendix B Classification and Finite Element Formulation of Viscous Flow Problems
- Appendix C Numerical Integration
- Appendix D Finite Element-Based Perturbation Analysis: Formulation of the Zeroth-Order Flow Field
- Appendix E Displaced-Rotor Operation: Perturbation Analysis
- Appendix F Rigorous Adaptation to Compressible-Flow Problems
- Index
2 - Some Methods for Solving Continuum Problems
Published online by Cambridge University Press: 05 June 2014
- Frontmatter
- Brief Contents
- Contents
- Preface
- 1 The Finite Element Method: Introductory Remarks
- 2 Some Methods for Solving Continuum Problems
- 3 Variational Approach
- 4 Requirements for the Interpolation Functions
- 5 Heat Transfer Applications
- 6 One-Dimensional Steady-State Problems
- 7 The Two-Dimensional Heat-Conduction Problem
- 8 Three-Dimensional Heat-Conduction Applications with Convection and Internal Heat Absorption
- 9 One-Dimensional Transient Problems
- 10 Fluid Mechanics Finite Element Applications
- 11 Use of Nodeless Degrees of Freedom
- 12 Finite Element Analysis in Curvilinear Coordinate
- 13 Finite Element Modeling of Flow in Annular Axisymmetric Passages
- 14 Extracting the Finite Element Domain from a Larger Flow System
- 15 Finite Element Application to Unsteady Flow Problems
- 16 Finite Element-Based Perturbation Approach to Unsteady Flow Problems
- Appendix A Natural Coordinates for Three-Dimensional Surface Elements
- Appendix B Classification and Finite Element Formulation of Viscous Flow Problems
- Appendix C Numerical Integration
- Appendix D Finite Element-Based Perturbation Analysis: Formulation of the Zeroth-Order Flow Field
- Appendix E Displaced-Rotor Operation: Perturbation Analysis
- Appendix F Rigorous Adaptation to Compressible-Flow Problems
- Index
Summary
Overview
There are many approaches to the solution of linear and nonlinear boundary-value problems, and they range from completely analytical to completely numerical. Of these, the following deserve attention:
• Direct integration (exact solution):
• Separation of variables
• Similarity solutions
• Fourier and Laplace transformations
• Approximate solutions
• Perturbation
• Power series
• Probability schemes
• Method of weighted residuals (MWR)
• Finite difference techniques
• Ritz method
• Finite element method
For a few problems, it is possible to obtain an exact solution by direct integration of the governing differential equation. This is accomplished, sometimes, by an obvious separation of variables or by applying a transformation that makes the variables separable and leads to a similarity solution. Occasionally, a Fourier or Laplace transformation of the differential equation leads to an exact solution. However, the number of problems with exact solutions is severely limited, and most of these have already been solved.
Because regular and singular perturbation methods are primarily applicable when the nonlinear terms in the equation are small in relation to the linear terms, their usefulness is limited. The power-series method is powerful and has been employed with some success, but because the method requires generation of a coefficient for each term in the series, it is relatively tedious. It is also difficult, if not impossible, to demonstrate that the series converges.
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- Publisher: Cambridge University PressPrint publication year: 2013