Basic Control Volume Finite Element Methods For Fluids And Solids

✍ Scribed by Vaughan R Voller

• Publisher: World Scientific Publishing Company
• Year: 2009
• Tongue: English
• Leaves: 185
• Series: Iisc Research Monographs
• Category: Library

✦ Synopsis

The Control Volume Finite Element Method (CVFEM) is a hybrid numerical method, combining the physics intuition of Control Volume Methods with the geometric flexibility of Finite Element Methods. The concept of this monograph is to introduce a common framework for the CVFEM solution so that it can be applied to both fluid flow and solid mechanics problems. To emphasize the essential ingredients, discussion focuses on the application to problems in two-dimensional domains which are discretized with linear-triangular meshes. This allows for a straightforward provision of the key information required to fully construct working CVFEM solutions of basic fluid flow and solid mechanics problems.

Contents: Governing Equations; The Essential Ingredients in a Numerical Solution; Control Volume Finite Element Data Structure; Control Volume Finite Element Method (CVFEM) Discretization and Solution; The Control Volume Finite Difference Method; Analytical and CVFEM Solutions of Advection-Diffusion Equations; A Plane Stress CVFEM Solution; CVFEM Stream Function-Vorticity Solution for a Lid Driven Cavity Flow; Notes Toward the Development of a 3-D CVFEM Code.

✦ Table of Contents

Contents......Page 12
Series Preface......Page 6
Preface......Page 10
1.1 Overview......Page 16
1.2 Objective and Philosophy......Page 17
1.3 The Basic Control Volume Concept......Page 18
1.4 Main Topics Covered......Page 19
2.1.1 Conservation of mass......Page 22
2.1.2 Conservation of linear momentum......Page 23
2.2 Specific Governing Equations......Page 24
2.2.2 Advection-diffusion of a scalar......Page 25
2.2.3 Stress and strain in an elastic solid......Page 26
2.2.4 Plane stress......Page 29
2.2.5 Plane strain......Page 31
2.2.6 Relationship between plane stress and plane strain......Page 32
2.2.7 The Navier-Stokes equations......Page 33
2.2.8 The stream-function—vorticity formulation......Page 35
3.1 The Basic Idea......Page 37
3.2.2 Mesh......Page 38
3.3 The Element and the Interpolation Shape Functions......Page 40
3.4 Region of Support and Control Volume......Page 43
3.5 The Discrete Equation......Page 44
4.2 The Mesh......Page 47
4.3.1 The region of support......Page 48
4.4 The Discrete Equation......Page 50
4.5 Summary......Page 53
5.1 The Approach......Page 54
5.2 Preliminary Calculations......Page 56
5.3 Steady State Diffusion......Page 59
5.4 Steady State Advection-Diffusion......Page 61
5.5.1 Volume source terms......Page 64
5.5.2 Source linearization......Page 65
5.5.3 Line source......Page 66
5.6 Coding Issues......Page 67
5.7.1 Face area calculations......Page 70
5.7.2 Convective condition......Page 72
5.7.3 Generalization of the convective boundary condition......Page 73
5.8 Solution......Page 74
5.9.1 A conjugate problem......Page 77
5.9.2 Diffusivity a function of field variable......Page 78
5.10 Transients......Page 80
5.11 Summary......Page 83
6.2 CVFDM Data Structure......Page 84
6.3 Coefficients and Sources......Page 86
6.4.2 Fixed value boundary......Page 89
6.5 Summary......Page 90
7.1 The Task......Page 91
7.2 Choice of Test Problems......Page 92
7.3 One-Dimensional Steady State Diffusion in a Finite Domain......Page 95
7.4 One-Dimensional Transient Diffusion in a Semi-Infinite Domain......Page 97
7.5 One-dimensional Transient Advection-Diffusion in a Semi-Infinite Domain......Page 100
7.6.1 Constant diffusivity......Page 102
7.6.2 Variable diffusivity and source term......Page 105
7.7.1 Constant diffusivity......Page 107
7.7.2 Variable diffusivity......Page 109
7.8.1 Problem......Page 110
7.8.2 Unstructured mesh solutions......Page 112
7.8.3 Structured mesh solutions......Page 113
7.9 The Recharge Well Problem......Page 115
8.2 The Stress Concentration Problem......Page 121
8.3 CVFEM Displacement Solution......Page 122
8.3.1 The CVFEM discrete equations......Page 124
8.3.2 Boundary conditions......Page 126
8.4 The Stress Solution......Page 127
8.4.2 Estimation of a nodal derivative......Page 130
8.4.3 Estimation of the nodal stress field......Page 134
8.5 Summary......Page 135
9.1 Introduction......Page 136
9.2 The Governing Equations......Page 137
9.3.1 Diffusion contributions......Page 138
9.3.2 Source terms......Page 140
9.4.1 Diffusion contributions......Page 141
9.4.2 The advection coefficients......Page 142
9.4.3 Boundary conditions......Page 143
9.5.1 Nested iteration......Page 144
9.5.2 Calculating the nodal velocity field......Page 145
9.6 Results......Page 147
10.1 The Tetrahedron Element......Page 153
10.2 Creating a Mesh of Tetrahedron Elements......Page 154
10.3 Geometric Features of Tetrahedrons......Page 158
10.4 Volume Shape Functions......Page 159
10.5 The Control Volume and Face......Page 161
10.6.1 Diffusive flux......Page 163
10.7 Summary......Page 164
Appendix A. A Meshing Code......Page 165
Appendix B. A CVFEM Code......Page 173
Bibliography......Page 182
Index......Page 184

✦ Subjects

Механика;Механика жидкостей и газов;Гидрогазодинамика;