Gradient Smoothing Methods with Programming. Applications to Fluids and Landslides

Contents
About the Authors
1. Introduction
	1.1 On numerical simulation
	1.2 Computational methods
	1.3 From physics to mathematics
		1.3.1 Well-posed physical problems
		1.3.2 Differential to algebra
		1.3.3 Methods and types of PDEs
	1.4 Types of numerical methods
		1.4.1 Strong-form vs weak-form governing equations
		1.4.2 Lagrangian methods vs Eulerian methods
		1.4.3 Grid-based methods vs meshfree methods
	1.5 Meshfree particle methods
		1.5.1 Concepts of meshfree particle methods
		1.5.2 A brief theory on SPH
		1.5.3 Pros and Cons of the SPH method
	1.6 Gradient smoothing methods
		1.6.1 Brief on GSM
		1.6.2 Brief on L-GSM
	1.7 Notation
	References
2. Theory for Gradient Smoothing Methods
	2.1 Constant-weighted gradient smoothing
		2.1.1 Gradient approximation
		2.1.2 Gradient smoothing operator with constant smoothing function
		2.1.3 Construction of gradient smoothing domains
		2.1.4 Spatial approximation schemes
			2.1.4.1 One-point quadrature schemes
		2.1.5 Effect of spatial approximation on solution accuracy
		2.1.6 Stencil analysis
			2.1.6.1 Basic principles for stencil assessment
			2.1.6.2 Stencils for approximated gradients
			2.1.6.3 Stencils for approximated Laplace operator
			2.1.6.4 Truncation errors
	2.2 Linearly-weighted gradient smoothing
		2.2.1 Construction of piecewise linear smoothing functions
		2.2.2 Linear-weighted gradient smoothing operator
			2.2.2.1 Determination of smoothing functions
		2.2.3 Approximation of spatial derivatives
			2.2.3.1 First-order derivatives
			2.2.3.2 Second-order derivatives
	2.3 Relationships between GSM and LW-GSM
		2.3.1 For interior nodes
		2.3.2 For boundary nodes
	2.4 A note on higher-order GSMs
	References
3. Eulerian GSM for Solving Poisson Equation
	3.1 Problem description
	3.2 Evaluation of numerical errors
	3.3 Cell types used in study
	3.4 GSM solutions
		3.4.1 The role of directional correction
		3.4.2 Comparison of four favorable schemes
		3.4.3 Robustness to irregularity of cells
	3.5 LW-GSM solutions
		3.5.1 Second Poisson problem
		3.5.2 Third Poisson problem
	3.6 Remarks
	References
4. Eulerian GSM for Compressible Flows
	4.1 Setting of the problem
		4.1.1 Governing equations for compressible flows
		4.1.2 Turbulence model
	4.2 GSM for compressible Navier–Stokes equations
		4.2.1 Discretization of governing equations
		4.2.2 Boundary conditions
		4.2.3 Time integration of GSM
	4.3 Numerical examples
		4.3.1 Case I: Inviscid flow over the NACA0012 airfoil
		4.3.2 Case II: Laminar flow over the flat plate
		4.3.3 Case III: Turbulent flow over the RAE2822 airfoil
		4.3.4 Case IV: Time-dependent compressible laminar flow over a circular cylinder
	References
5. Eulerian GSM for Incompressible Flows
	5.1 Setting of the problem
		5.1.1 Governing equations for incompressible flows
	5.2 GSM for incompressible flows
		5.2.1 Discretization of governing equations
		5.2.2 Boundary conditions
	5.3 Numerical examples
		5.3.1 Steady-state flow past a circular cylinder
		5.3.2 Steady-state flow over a backstep
		5.3.3 Steady-state lid-driven cavity flow
		5.3.4 Time-dependent flow over a circular cylinder
	5.4 Concluding remarks
	References
6. Theory and Formulation for Lagrangian GSM
	6.1 Assumptions used in L-GSM
	6.2 Gradient smoothing for L-GSM
		6.2.1 For 2D problems
		6.2.2 For 3D problems
		6.2.3 Accuracy analysis of the smoothed gradient
	6.3 Free surface treatments for L-GSM
		6.3.1 Consistent form of gradient smoothing on free surface
		6.3.2 Normalized form of gradient smoothing
	6.4 Supporting particle selections
		6.4.1 Global mesh and re-mesh
		6.4.2 Local supporting particle selection algorithm
		6.4.3 Remarks
	6.5 Construction of smoothing domain
		6.5.1 For 2D problems
		6.5.2 For 3D problems
	6.6 Numerical oscillation and artificial viscosity
	6.7 On conservation of L-GSM
	6.8 Boundary treatments
		6.8.1 Free-slip solid boundary treatments
		6.8.2 No-slip solid boundary treatment
		6.8.3 Periodical boundary treatment
	6.9 Time integration
	6.10 Accuracy condition of L-GSM
		6.10.1 Numerical tests on accuracy of smoothed gradient
		6.10.2 Accuracy comparison of L-GSM and SPH gradient operators
		6.10.3 Accuracy of 3D smoothed gradient in local L-GSM
		6.10.4 Remarks
	6.11 Stability condition of L-GSM
	References
7. L-GSM for Incompressible Hydrodynamics
	7.1 Governing equations
	7.2 Numerical examples
		7.2.1 Couette flow
		7.2.2 Poiseuille flow
		7.2.3 Shear-driven cavity
		7.2.4 Dam break
		7.2.5 Wall impact of breaking dam
		7.2.6 Water discharge
		7.2.7 Droplet splash
	References
8. L-GSM for Granular Flows in Geotechnical Engineering
	8.1 Constitutive and governing equations
	8.2 Numerical examples
		8.2.1 Soil column collapse
			8.2.1.1 Non-cohesive soil
			8.2.1.2 Cohesive soil
		8.2.2 Earthquake-induced landslide
			8.2.2.1 Daguangbao landslide
			8.2.2.2 Tangjiashan landslide
			8.2.2.3 Wangjiayan landslide
	8.3 Efficiency estimation
	8.4 Efficiency via numerical tests
	8.5 Concluding remarks
	References
9. Programming with GSMs and Source Codes
	9.1 Overview
	9.2 GSM gradient approximation
		9.2.1 2D cases
		9.2.2 3D cases
	9.3 GSM simulation of Poiseuille flow
	9.4 L-GSM simulation of Couette flow
Index