Voller Vaughan R. Basic Control Volume Finite Element Methods for Fluids and Solids

World Scientific Publishing Co. Pte. Ltd., 2009. XIV, 170 p. — ISBN13: 978-981-283-498-0, ISBN10: 981-283-498-2.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.Overview
Objective and Philosophy
The Basic Control Volume Concept
Main Topics CoveredGoverning EquationsThe Euler Equations of Motion
Conservation of mass
Conservation of linear momentum
Conservation of a scalar
Specific Governing Equations
Mass conservation in an incompressible flow
Advection-diffusion of a scalar
Stress and strain in an elastic solid
Plane stress
Plane strain
Relationship between plane stress and plane strain
The Navier-Stokes equations
The stream-function—vorticity formulationThe Essential Ingredients in a Numerical SolutionThe Basic Idea
The Discretization: Grid, Mesh, and Cloud
Grid
Mesh
Cloud
Discretizations for the Control Volume Finite
Element Method
The Element and the Interpolation Shape Functions
Region of Support and Control Volume
The Discrete EquationControl Volume Finite Element Data StructureThe Task
The Mesh
The Data Structure
The region of support
The boundary
The Discrete EquationControl Volume Finite Element Method (CVFEM)Discretization and Solution
The Approach
Preliminary Calculations
Steady State Diffusion
Steady State Advection-Diffusion
Steady State Advection-Diffusion with Source Terms
Volume source terms
Source linearization
Line source
Coding Issues
Boundary Conditions
Face area calculations
Convective condition
Generalization of the convective boundary condition
Solution
Handling Variable Diffusivity
A conjugate problem
Diffusivity a function of field variable
TransientsThe Control Volume Finite Difference MethodThe Task
CVFDM Data Structure
Coefficients and Sources
Boundary Conditions
Insulated (no-flow) boundary
Fixed value boundaryAnalytical and CVFEM Solutions of Advection-DiffusionEquations
The Task
Choice of Test Problems
One-Dimensional Steady State Diffusion in a Finite Domain
One-Dimensional Transient Diffusion in a Semi-Infinite
Domain
One-dimensional Transient Advection-Diffusion in a
Semi-Infinite Domain
Steady State Diffusion in an Annulus
Constant diffusivity
Variable diffusivity and source term
Steady State Advection Diffusion in an Annulus
Constant diffusivity
Variable diffusivity
Transient Diffusion from a Line Source
Problem
Unstructured mesh solutions
Structured mesh solutions
The Recharge Well ProblemA Plane Stress CVFEM SolutionThe Stress Concentration Problem
CVFEM Displacement Solution
The CVFEM discrete equations
Boundary conditions
Solution
The Stress Solution
Stress in an element
Estimation of a nodal derivative
Estimation of the nodal stress fieldCVFEM Stream function-Vorticity Solution for a Lid Driven Cavity FlowThe Governing Equations
The CVFEM Discretization of the Stream Function Equation
Diffusion contributions
Source terms
Boundary conditions
The CVFEM Discretization of the Vorticity Equation
Diffusion contributions
The advection coefficients
Boundary conditions
Solution Steps
Nested iteration
Calculating the nodal velocity field
ResultsNotes toward the Development of a 3-D CVFEM CodeThe Tetrahedron Element
Creating a Mesh of Tetrahedron Elements
Geometric Features of Tetrahedrons
Volume Shape Functions
The Control Volume and Face
Approximation of Face Fluxes
Diffusive flux
Advective fluxAppendix A. A Meshing CodeAppendix B. A CVFEM Code