Linear and nonlinear ultrasound simulations using the discontinuous Galerkin method

Area of Science:

• Computational physics
• Acoustics
• Numerical analysis

Background:

• Simulating transient ultrasound propagation requires methods that handle nonlinear acoustic wave equations.
• Discontinuous Galerkin (DG) methods offer high-order accuracy, geometric flexibility, and scalability, making them suitable for complex wave propagation problems.

Purpose of the Study:

• To develop and validate a nodal discontinuous Galerkin (DG) code for simulating transient ultrasound propagation based on a nonlinear wave equation.
• To implement a dynamic sub-grid scale stabilization method to mitigate Gibbs oscillations in acoustic shock waves.

Main Methods:

• A nonlinear acoustic wave equation was formulated in a first-order flux form.
• The equation was discretized using the nodal discontinuous Galerkin (DG) method.
• A dynamic sub-grid scale stabilization technique was introduced to handle shock waves.

Main Results:

• A two-dimensional axisymmetric DG code was developed and tested.
• Numerical results from the DG code were compared with linear and Khokhlov-Zabolotskaya-Kuznetsov (KhKz)-based simulations.
• The DG simulations successfully captured nonlinearity, thermoviscous absorption, and diffraction for both flat and focused pistons.

Conclusions:

• The nodal DG method is an effective approach for simulating transient ultrasound propagation, accurately modeling nonlinear acoustic phenomena.
• The developed DG code provides a robust tool for analyzing complex ultrasound wave behaviors in homogeneous media.