This textbook comprehensively covers the fundamentals behind mathematical modeling of engineering problems to obtain the required solution.
It comprehensively discusses modeling concepts through conservation principles with a proper blending of mathematical expressions. The text discusses the basics of governing equations in algebraic and differential forms and examines the importance of mathematics as a tool in modeling. It covers important topics including modeling of heat transfer problems, modeling of flow problems, modeling advection-diffusion problems and Navier-Stokes equations in depth. Pedagogical features including solved problems and unsolved exercises are interspersed throughout the text for better understanding.
The textbook is primarily written for senior undergraduate and graduate students in the field of mechanical engineering for courses on modeling and simulation. The textbook will be accompanied by teaching resource including a solution manual for the instructors.
Author(s): Krishnan Murugesan
Publisher: CRC Press
Year: 2022
Language: English
Pages: 370
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Author Biography
Chapter 1 Introduction
1.1 Modeling
1.1.1 Physical Models
1.1.2 Mathematical Models
1.1.2.1 Perfect Gas Equation
1.1.2.2 Hooke’s Law
1.1.2.3 Deflection of Beam under Load
1.1.2.4 Fluid Mechanics
1.1.2.5 Heat Transfer
1.2 Simulation
1.3 Conservation Principles
1.3.1 Mass Conservation
1.3.2 Momentum Conservation
1.3.3 Energy Conservation
1.3.4 Species Conservation
1.4 Types of Physical Problems
1.4.1 Equilibrium Problems
1.4.2 Eigen Value Problems
1.4.3 Propagation Problems
1.5 Models in Engineering Analysis
1.5.1 Lumped Parameter Model
1.5.2 Continuum Based Model
1.6 Solution of Differential Equations
1.6.1 Analytical Techniques
1.6.2 Numerical Techniques
1.6.3 Computing Techniques
References
Exercise Problems
Quiz Questions
Chapter 2 Conservation Equations
2.1 Solid Medium
2.1.1 Energy Transport in Unsteady State Conditions
2.1.1.1 Generalized Conduction Equation in Cartesian Coordinates
2.1.1.2 Generalized Conduction Equation in Cylindrical Coordinates
2.1.1.3 Generalized Conduction Equation in Spherical Coordinates
2.1.1.4 Initial and Boundary Conditions
2.1.1.5 Initial Condition
2.1.1.6 Boundary Conditions
2.1.2 Energy Transport in Steady State Condition
2.1.2.1 Steady State Heat Conduction in Plane Wall
2.1.2.2 Steady State Heat Conduction in Cylinder
2.1.2.3 Steady State Heat Conduction in Sphere
2.2 Fluid Medium
2.2.1 Mass Conservation
2.2.1.1 Material Derivative Form
2.2.1.2 Incompressible Fluid Flow
2.2.2 Momentum Conservation
2.2.2.1 Relation between Stress and Viscosity
2.2.2.2 Momentum Balance Equations for Incompressible Flow (μ=constant)
2.2.3 Energy Conservation
2.2.3.1 Energy Balance
2.2.3.2 Rate of Change of Energy in CV
2.2.3.3 Net Efflux of Energy from CV
2.2.3.4 Rate of Work Done by Surface Forces
2.2.3.5 Work Done by Body Forces
2.2.3.6 Net Addition of Heat due to Conduction and Radiation Heat Transfer
2.2.3.7 Heat Generation within Control Volume
2.2.4 Species Conservation
References
Exercise Problems
Quiz Questions
Chapter 3 Finite Difference and Finite Volume Methods
3.1 Finite Difference Method
3.1.1 One-Dimensional Conduction
3.1.2 Taylor’s Series Principle
3.1.3 Polynomial Method
3.1.4 Application to Ordinary Differential Equations
3.1.4.1 Equations for the Boundary Nodes 1 and M
3.1.5 Application to Partial Differential Equations
3.1.5.2 Difference Equations for Boundary Conditions
3.1.5.3 Corner Nodes
3.1.5.4 Boundary Nodes
3.1.5.5 Comparison of Two-Dimensional Conduction Results with Analytical Solution
3.2 Finite Volume Method
3.2.1 Heat Flux Boundary Condition at M (x=L)
3.2.2 Convective Boundary Condition at Node M (x=L)
3.2.3 Example Problem for Finite Volume Method – Fin
3.2.4 One-Dimensional and Two-Dimensional Applications
3.2.4.2 Two-Dimensional Application
3.2.4.3 Boundary Conditions
3.2.4.4 Corner Nodes
3.2.5 Complex Geometry and Variable Property
3.2.5.1 Complex Geometry
3.2.5.2 Variable Property
3.2.5.3 Variable Area
References
Exercise Problems
Quiz Questions
Chapter 4 Finite Element Method
4.1 Galerkin’s Weighted Residual Method
4.1.1 Integration of Shape Functions
4.1.2 Boundary Conditions
4.1.2.1 Convective Boundary Condition
4.1.2.2 Dirichlet Boundary Condition
4.1.3 Example Problem: Fin (Computer Code fin_FEM.for)
4.2 Domain Discretization and Isoparametric Formulation
4.2.2 Isoparametric Formulation
4.3 Discretization of One-Dimensional Domain
4.4 Discretization of Two-Dimensional Domain
4.4.1 Rectangular and Quadrilateral Elements
4.5 Discretization of Three-Dimensional Domain
4.6 Mesh Generation
4.7 Transfinite Interpolation Technique (TFI)
4.7.1 Multi-Block TFI Grid Generation
4.7.2 Three-Dimensional TFI Meshing
4.8 Time-Dependent Problems
4.8.1 Stability Conditions
4.8.1.1 Explicit Scheme
4.8.1.2 Implicit Scheme
4.8.1.3 Semi-Implicit Scheme (Crank-Nicholson Scheme)
4.8.1.4 Significance of Fourier Number
4.8.1.5 Alternate Direction Implicit (ADI) Method
References
Exercise Problems
Quiz Questions
Chapter 5 Modeling of Heat Transfer Problems
5.1 Heat Transfer Problem – One-Dimensional Conduction with Heat Generation
5.1.1 Derivation of Energy Conservation Equation
5.1.2 Identification of Boundary Conditions
5.1.3 Solution Using Finite Element Method
5.1.4 Incorporation of Boundary Condition
5.1.5 Computational Algorithm
5.1.6 Computer Programming
5.1.7 Mesh Sensitivity and Validation Results
5.1.8 Simulation Parameters and Results
5.2 Two-Dimensional Problem – Heat and Mass Transfer through Soil: Landmine Detection
5.2.1 Derivation of Conservation Equations
5.2.1.1 Conservation Equation for Heat and Moisture Transport within the Soil Medium
5.2.2 Initial and Boundary Conditions
5.2.2.2 Boundary Conditions – Soil Medium
5.2.2.3 Top Side
5.2.2.4 Bottom Side
5.2.2.5 Vertical Sides
5.2.2.6 Mine-Soil Interface Boundary Condition
5.2.3 Solution Using Finite Element Method
5.2.4 Inclusion of Convective Boundary Condition on the Top Surface of the Soil Medium
5.2.5 Solution of Energy Equation for the Landmine
5.2.6 Computational Algorithm
5.2.7 Computer Programming
5.2.8 Simulation Parameters and Results
5.2.9 Discussion of Simulation Results
5.2.9.1 Mesh Sensitivity and Validation Results [10]
5.2.9.2 Simulation Results [10]
Effect of Depth and Size of Mine
Effect of Moisture Content of Soil Medium
Temperature Contours of Soil Medium with Mine at Different Depths
References
Exercise Problems
Quiz Questions
Chapter 6 Modeling of Flow Problems
6.1 Fluid Mechanics – Filling of Water Tank
6.1.1 Derivation of Mass and Momentum Conservation Equations
6.1.2 Boundary Conditions and Initial Conditions
6.1.3 Solution Using Analytical Method
6.1.4 Computational Algorithm and Computer Program
6.1.5 Simulation Parameters and Discussion of Results
6.2 Two-Dimensional Flow Problems – Stokes Flow
6.2.1 Description of Problem
6.2.2 Mathematical Modeling
6.3 Three-Dimensional Stokes Flow
6.3.1 Governing Equations for Three-Dimensional Stokes Flow
Vorticity Transport Equations
Velocity Poisson Equations
Boundary Conditions
6.3.2 Finite Element Solution Procedure
6.3.3 Enforcement of Dirichlet Boundary Conditions in Finite Element Solution Procedure
6.3.3.1 Computational Steps to Incorporate Dirichlet Boundary Conditions
6.3.4 Global Matrix-Free Finite Element Algorithm
6.3.4.1 Matrix Storage Schemes for Large Size Problems and Solvers
6.3.4.2 BICGSTAB and Element-by-Element Scheme for Parallel Computing
6.3.4.3 Procedure to Implement Global Matrix-Free Finite Element Algorithm
6.4 Results for Three-Dimensional Stokes Flow
6.4.1 Comparison of Memory Storage of GMFFE Algorithm with Column Format Scheme
6.4.2 Flow Results for Three-Dimensional Stokes Flow Using 513 Mesh
6.4.2.2 Velocity Vectors Distribution
References
Exercise Problems
Quiz Questions
Chapter 7 Navier-Stokes Equations
7.1 Momentum Balance of Fluid in a System
7.1.1 Fluid Dynamics
7.2 Navier-Stokes Equations in Primitive Variables Form
7.2.1 Navier-Stokes Equations
7.2.2 Application of Predictor-Corrector Method
7.2.3 Finite Element Solution Procedure
7.2.4 Computational Algorithm
7.2.4.1 Computer Program – Subroutines
7.3 Navier-Stokes Equations in Velocity-Vorticity Form
7.3.1 Derivation of Velocity-Vorticity Equations as Generalized Formulation
Body Force Terms
7.3.2 Computation of Vorticity Boundary Conditions
7.3.2.1 Node i on Side AB – For Wall Normal Parallel to Positive y-Axis
7.3.2.2 Node j on Side CD – For Wall Normal Parallel to Negative y-Axis
7.3.2.3 Node k on Side DA – For Wall Normal Parallel to Positive x-Axis
7.3.2.4 Node m on Side BC – For Wall Normal Parallel to Negative x-Axis
7.3.3 Solution Using Finite Element Method
7.3.4 Finite Element Formulation of Vorticity Transport Equation
7.3.5 Finite Element Solution Procedure for Velocity Poisson Equations
7.3.6 Computational Algorithm
7.3.7 Simulation of Lid-Driven Square Cavity Flow Problem
7.3.8 Simulation Results
7.3.9 Simulation of Natural Convection in a Square Cavity
7.3.9.1 Finite Element Solution Procedure
7.3.9.2 Simulation Results for Natural Convection in a Differentially Heated Square Cavity
References
Exercise Problems
Quiz Questions
Index
