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Table of contents
1 Introduction
1.1 Motivation and Objective
1.2 Popular Models and Numerical Methods for Propagation of Acoustical Shock Waves
1.3 Numerical Methods for Complex Geometry and Acoustical Shock Waves
1.3.1 Choice of the Method
1.3.2 Shock Management
1.4 Outline of the Manuscript
2 Equations of Propagation in Nonlinear Acoustics
2.1 Conservation Laws
2.2 Equations for Nonlinear Acoustics
2.3 Dimensionless Formulation of the System of Equations
2.3.1 Characteristic Parameters and Variables
2.4 Summary
2.5 Comparison with other Equations of Nonlinear Acoustics
2.5.1 Conservative to Primitive form
2.5.2 Kuznetsov Equation
2.5.3 Westervelt Equation
2.5.4 KZ Equation
2.5.5 Inviscid Burgers Equation
2.6 Conclusions
3 Discontinuous Galerkin Method
3.1 Nodal Discontinuous Galerkin Method in 1D
3.1.1 Weak Formulation
3.1.2 Computations in Reference Element
3.1.3 Assembling
3.2 Nodal Discontinuous Galerkin Method in 2D
3.2.1 Weak Formulation
3.2.2 Computations in Reference Element
3.2.3 Assembling
3.3 Brief Review on GPU Implementation
3.4 Time Discretization
3.5 Application 1D: Advection Equation
3.6 Conclusions
4 Shock Management in One-Dimension
4.1 Illustration
4.2 Slope Limiters
4.2.1 Slope Limiter: Cockburn
4.2.2 Slope Limiter: Biswas
4.2.3 Slope Limiter: Burbeau
4.2.4 Numerical Experiment
4.3 Method of Global Artificial Viscosity
4.3.1 Local Discontinuous Galerkin Method in 1D
4.3.2 Numerical Experiment
4.4 Element Centered Smooth Artificial Viscosity
4.4.1 Shock Sensor
4.4.2 Smooth Artificial Viscosity
4.4.3 Implementation Issues
4.5 Validation
4.5.1 Inverted Sine-period to N-wave
4.5.2 Sine-period to Sawtooth
4.5.3 N-wave
4.5.4 Sawtooth
4.5.5 Multiple Shocks
4.6 Conclusions
5 Shock Management in Two-Dimensions
5.1 Equations of Nonlinear Acoustics
5.2 Convective-Diffusive System for Nonlinear Acoustics
5.3 Local Discontinuous Galerkin Implementation
5.3.1 Weak Formulation
5.3.2 Numerical Fluxes
5.3.3 Nodal Approximation
5.3.4 Assembling
5.4 Element Centered Smooth Artificial Viscosity
5.4.1 Shock Sensor
5.4.2 Smooth Artificial Viscosity
5.5 Numerical Explanation of the Shock Sensor
5.5.1 First-Order Contribution to the Shock Sensor
5.5.2 Highest-Order Contribution to the Shock Sensor
5.6 Implementation Issues and Validation
5.7 Conclusions
6 Applications
6.1 Reflection of Acoustical Shock Waves
6.1.1 Numerical Experiments
6.1.2 Results and Discussion
6.2 Focusing of continuous (shock) waves: application to HIFU
6.2.1 Mesh Renement Based on ECSAV
6.2.2 Low resolution simulation
6.2.3 Local high resolution mesh
6.2.4 Focusing in a homogeneous medium
6.2.5 Intensity near the focus
6.2.6 Focusing in a medium with an obstacle
6.2.7 Conclusions
7 Conclusions and Perspectives
References
