Numerical Solutions Applied to Heat Transfer with the SPH Method: A Verification of Approximations for Speed and Accuracy (SpringerBriefs in Mathematics)

Foreword
Preface
Acknowledgments
Contents
Acronyms
	Symbols
	Greek Symbols
	Subscript
	Superscript
List of Figures
List of Tables
1 Introduction
	Problem Characterization
	Relevance
	Objective
	Authors\' Suggestion
	Book Structure
	References
2 Numerical Modeling of Heat Diffusion
	2.1 Numerical Model
		2.1.1 SPH Formulation
		Integral Representation of a Function
		Particle Approximation
		2.1.2 Kernel Function Properties
		SPH Kernel Function: Definition and Models
		Classic Models of SPH Kernel Functions and Their Derivatives (Ww*γ(ϕ))
	2.2 Boundary Treatment
		Ghost Particles
		Dummy Particles
	2.3 Canonical Discretization with Fixed Sensor Particle
		Particle Recruitment
	2.4 Discretization of the Steady State Heat Diffusion Equation
	2.5 Discretization of the Unsteady State Heat Diffusion Equation
		2.5.1 Crank–Nicolson Method for SPH
	2.6 Solvers
		TDMA Solver
		PDMA Solver
		Compressed Sparse Row (CSR)
		Gauss-Seidel-S CSR Solver
	2.7 Multigrid and Multilevel
		Go to Inconsistent Sublevel
		Return to Consistent Sublevel
		Gauss-Seidel Solver
	2.8 Numerical Error
		Discretization Error Estimates a priori
		Discretization Error Estimates a posteriori
		Truncation Error
		Iteration Error
		Round-Off Error
		Numerical Pollution Error
		Programming Error
	2.9 Verification of Numerical Solutions
		Global Variables
		Average Temperature
		Vector Norms
		Matrix Norms
	2.10 Repeated Richardson Extrapolation
	References
3 Numerical Error Analysis and Heat Diffusion Models
	3.1 Mathematical Models
		3.1.1 Steady State Heat Diffusion Equation
		3.1.2 Unsteady State Heat Diffusion Equation
	3.2 Numerical Models
		3.2.1 Computational Domain
		3.2.2 Coding of Numerical Examples
		3.2.3 C1–1Ds Analytic Model
		3.2.4 C2–1Ds Analytic Model
		3.2.5 C3–1Ds Analytic Model
		3.2.6 C1–1Du Analytic Model
		3.2.7 C1–2Ds Analytic Model
		3.2.8 C2–2Ds Analytic Model
	3.3 Variables of Interest
	3.4 SPH Sources of Numerical Errors
		3.4.1 A priori Discretization Error Estimates
		3.4.2 A posteriori Discretization Error Estimates
		3.4.3 Local Truncation Error
			3.4.3.1 Discretization of the 1D Laplacian Operator
			3.4.3.2 Asymmetric Approximation for First Derivative One-Dimensional
			3.4.3.3 Discretization of the 2D Laplacian Operator
		3.4.4 Iteration Error
		3.4.5 Round-Off Error
		3.4.6 Particle Inconsistency Error
			3.4.6.1 Particle Inconsistency Error in 1D Models
			3.4.6.2 Particle Inconsistency Error in 2D Models
		3.4.7 Kernel Smoothing Error
		3.4.8 Numerical Pollution Error
	References
4 SPH Applied to Computational Heat Transfer Problems
	4.1 General Data from the Numerical Simulations
	4.2 Steady State Heat Diffusion Equation 1D
		4.2.1 C1-1Ds Numerical Model
			4.2.1.1 Qualitative Verification of Numerical Solutions for Local Variable
			4.2.1.2 Qualitative Verification of Numerical Solutions for Global Variable
		4.2.2 C2-1Ds Numerical Model
			4.2.2.1 Qualitative Verification of Numerical Solutions for Local Variable
			4.2.2.2 Qualitative Verification of Numerical Solutions for Global Variable
		4.2.3 C3-1Ds Numerical Model
			4.2.3.1 Qualitative Verification of Numerical Solutions for Local Variable
			4.2.3.2 Qualitative Verification of Numerical Solutions for Global Variable
	4.3 Unsteady State Heat Diffusion Equation 1D
		4.3.1 C1-1Du Numerical Model
			4.3.1.1 Qualitative Verification of Numerical Solutions for Local Variable
			4.3.1.2 Qualitative Verification of Numerical Solutions for Global Variable
	4.4 Steady State Heat Diffusion Equation 2D
		4.4.1 C1-2Ds Numerical Model
			4.4.1.1 Qualitative Verification of Numerical Solutions for Local Variable
			4.4.1.2 Qualitative Verification of Numerical Solutions for Global Variable
			4.4.1.3 Quantitative Verification of Numerical Solutions
		4.4.2 C2-2Ds Numerical Model
	4.5 Application
		4.5.1 Steady State Heat Diffusion Equation 2D Using Canonical Discretization with Fixed Sensor Particle
	References
5 Conclusion
	Reference