Finite Element Methods for Flow Problems
Taking an engineering rather than a mathematical bias, this valuable reference resource details the fundamentals of stabilised finite element methods for the analysis of steady and time-dependent fluid dynamics problems.
Format: Hardcover, English, 350 pages
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In recent years there have been significant developments in the development of stable and accurate finite element procedures for the numerical approximation of a wide range of fluid mechanics problems. Taking an engineering rather than a mathematical bias, this valuable reference resource details the fundamentals of stabilised finite element methods for the analysis of steady and time-dependent fluid dynamics problems. Organised into six chapters, this text combines theoretical aspects and practical applications and offers coverage of the latest research in several areas of computational fluid dynamics.
- Coverage includes new and advanced topics unavailable elsewhere in book form
- Collection in one volume of the widely dispersed literature reporting recent progress in this field
- Addresses the key problems and offers modern, practical solutions
Due to the balance between the concise explanation of the theory and the detailed description of modern practical applications, this text is suitable for a wide audience including academics, research centres and government agencies in aerospace, automotive and environmental engineering.
Chapter 1 provides an overview of the main issues encountered in the use of standard, Galerkin based, finite element methods for approximating problems governed by non-self adjoint equations and indicates how these deficiencies spurred the development of Petrov-Galerkin methods by which the optimal approximation property of the Galerkin method in self-adjoint cases can be carried over to problems governed by non-symmetric operators.
Chapter 2 addresses the finite element approximation of steady convection dominated transport problems. Starting from the first finite element methods of upwind type developed in the early 1970's, the reader is progressively introduced to a general class of Petrov-Galerkin methods possessing good accuracy and stability properties. Included here are the Streamline-Upwind Petrov-Galerkin (SUPG) method, the Galerkin/least-squares (GLS) method, as well as bubble function and wavelet-based methods.
Chapter 3 is then devoted to the presentation of finite element based algorithms specifically adapted for approximating unsteady transport problems describing pure convection. Some of the most widely used characteristic-based finite element methods are reviewed in the first part of the chapter. Then, the attention is focussed on methods involving direct time integration of the semi-discrete equations resulting from a finite element spatial discretization. Emphasis is placed on new methodologies, such as the Taylor-Galerkin method and the Pade/least-squares method, which provide time-accurate and stable approximations to convective transport problems.
Chapter 4 extends the above generalised finite element methods for approximating mixed problems describing transport by convection and diffusion. Several methods are described which preserve the possibility of using simple finite elements in the presence of diffusion operators. This includes methods based on operator-splitting techniques, on the reduction of the second-order convection-diffusion equation to a first-order system, and on the introduction of suitable assumptions into the Galerkin/least-squares method.
Chapter 5 is devoted to the finite element solution of viscous incompressible flow problems in the laminar regime. Here, the main difficulties are the incompressibility condition, and consequently the pressure computation, the advective-diffusive character of the Navier-Stokes equations, which requires appropriate stabilising techniques, and the nonlinearity of the equations. Emphasis is placed on the use of fractional-step finite element methods and applications include natural convection problems.
Chapter 6 covers the finite element solution of inviscid compressible flow problems governed by the Euler equations of gas dynamics and the numerical simulation of coupled fluid-structure problems formulated in the Arbitrary Lagrangian-Eulerian (ALE) kinematical description. Here, the attention is focused on a fractional-step approach to the time integration of the governing conservation equations in ALE form, on the implementation of high-resolution shock-capturing schemes in a finite element framework, and on the ALE treatment of fluid-structure interaction.