Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams.

On the basis of recently developed Fourier continuation (FC) methods and associated efficient parallelization techniques, this text introduces numerical algorithms that, due to very low dispersive errors, can accurately and efficiently solve the types of nonlinear partial differential equation (PDE) models of nonlinear acoustics in hundred-wavelength domains as arise in the simulation of focused medical ultrasound. As demonstrated in the examples presented in this text, the FC approach can be used to produce solutions to nonlinear acoustics PDEs models with significantly reduced discretization requirements over those associated with finite-difference, finite-element and finite-volume methods, especially in cases involving waves that travel distances that are orders of magnitude longer than their respective wavelengths. In these examples, the FC methodology is shown to lead to improvements in computing times by factors of hundreds and even thousands over those required by the standard approaches. A variety of one-and two-dimensional examples presented in this text demonstrate the power and capabilities of the proposed methodology, including an example containing a number of scattering centers and nonlinear multiple-scattering events.

[1]  Thomas Dreyer,et al.  Full-wave modeling of therapeutic ultrasound: nonlinear ultrasound propagation in ideal fluids. , 2002, The Journal of the Acoustical Society of America.

[2]  D. Borup,et al.  Simulation of acoustic wave propagation in dispersive media with relaxation losses by using FDTD method with PML absorbing boundary condition , 1999, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[3]  Robin O Cleveland,et al.  Modeling the propagation of nonlinear three-dimensional acoustic beams in inhomogeneous media. , 2007, The Journal of the Acoustical Society of America.

[4]  Oscar P. Bruno,et al.  Fast, High-Order, High-Frequency Integral Methods for Computational Acoustics and Electromagnetics , 2003 .

[5]  B. Gustafsson.  High Order Difference Methods for Time Dependent PDE , 2008 .

[6]  N. O. Weiss,et al.  Convective difference schemes , 1966 .

[7]  Lawrence A Crum,et al.  Acoustic characterization of high intensity focused ultrasound fields: a combined measurement and modeling approach. , 2008, The Journal of the Acoustical Society of America.

[8]  Thomas Dreyer,et al.  Full wave modeling of therapeutic ultrasound: efficient time-domain implementation of the frequency power-law attenuation. , 2004, The Journal of the Acoustical Society of America.

[9]  Graham Ashcroft,et al.  Optimized prefactored compact schemes , 2003 .

[10]  Oscar P. Bruno,et al.  Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis , 2007, J. Comput. Phys..

[11]  Yun Jing,et al.  Verification of the westervelt equation for focused transducers , 2011, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[12]  K W A van Dongen,et al.  A contrast source method for nonlinear acoustic wave fields in media with spatially inhomogeneous attenuation. , 2011, The Journal of the Acoustical Society of America.

[13]  Jing Gong,et al.  A stable and conservative high order multi-block method for the compressible Navier-Stokes equations , 2009, J. Comput. Phys..

[14]  T. Varslot,et al.  Forward propagation of acoustic pressure pulses in 3D soft biological tissue , 2006 .

[15]  Vera A. Khokhlova,et al.  A modification of the spectral description of nonlinear acoustic waves with discontinuities , 1996 .

[16]  David I. Gottlieb,et al.  The Theoretical Accuracy of Runge-Kutta Time Discretizations for the Initial Boundary Value Problem: A Study of the Boundary Error , 1995, SIAM J. Sci. Comput..

[17]  J. Boyd.  A Comparison of Numerical Algorithms for Fourier Extension of the First, Second, and Third Kinds , 2002 .

[18]  Miguel R. Visbal,et al.  On the use of higher-order finite-difference schemes on curvilinear and deforming meshes , 2002 .

[19]  Leland Jameson.  High Order Schemes for Resolving Waves: Number of Points per Wavelength , 2000, J. Sci. Comput..

[20]  R. Waag,et al.  An equation for acoustic propagation in inhomogeneous media with relaxation losses , 1989 .

[21]  Numerical Solution of the Kzk Equation for Pulsed Finite Amplitude Sound Beams in Thermoviscous Fluids , 1993 .

[22]  E. Zabolotskaya,et al.  Quasi-plane waves in the nonlinear acoustics of confined beams , 1969 .

[23]  H. O'neil.  Theory of Focusing Radiators , 1949 .

[24]  M. Hamilton,et al.  Time‐domain modeling of pulsed finite‐amplitude sound beams , 1995 .

[25]  Nathan Albin,et al.  A spectral FC solver for the compressible Navier-Stokes equations in general domains I: Explicit time-stepping , 2011, J. Comput. Phys..

[26]  François Coulouvrat,et al.  Acoustic shock wave propagation in a heterogeneous medium: a numerical simulation beyond the parabolic approximation. , 2011, The Journal of the Acoustical Society of America.

[27]  K. Hynynen,et al.  Bio-acoustic thermal lensing and nonlinear propagation in focused ultrasound surgery using large focal spots: a parametric study. , 2002, Physics in medicine and biology.

[28]  Mark Lyon,et al.  High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations , 2010, J. Comput. Phys..

[29]  Nathan Albin,et al.  Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws , 2011, J. Comput. Phys..

[30]  David I. Gottlieb,et al.  On the Removal of Boundary Errors Caused by Runge-Kutta Integration of Nonlinear Partial Differential Equations , 1994, SIAM J. Sci. Comput..

[31]  William D. Henshaw,et al.  Parallel computation of three-dimensional flows using overlapping grids with adaptive mesh refinement , 2008, J. Comput. Phys..

[32]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[33]  William D. Henshaw,et al.  A High-Order Accurate Parallel Solver for Maxwell's Equations on Overlapping Grids , 2005, SIAM J. Sci. Comput..

[34]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[35]  L. Fox.  Finite Difference Methods for Hyperbolic Equations , 1962 .

[36]  Yun Jing,et al.  On the use of Gegenbauer reconstructions for shock wave propagation modeling , 2010, 2010 IEEE International Ultrasonics Symposium.

[37]  Yoichiro Matsumoto,et al.  Development of high intensity focused ultrasound simulator for large‐scale computing , 2011 .

[38]  G. Trahey,et al.  A heterogeneous nonlinear attenuating full- wave model of ultrasound , 2009, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[39]  R. M. Alford,et al.  ACCURACY OF FINITE‐DIFFERENCE MODELING OF THE ACOUSTIC WAVE EQUATION , 1974 .

[40]  Algorithm for the nonlinear propagation of acoustic beams from phased arrays and nonplanar sources , 2006, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[41]  P. Morgan,et al.  A Parallel Overset Grid High-Order Flow Solver for Large Eddy Simulation , 2006, J. Sci. Comput..

[42]  Mark Lyon,et al.  High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements , 2010, J. Comput. Phys..

[43]  Nassir Navab,et al.  Fast Ultrasound Image Simulation Using the Westervelt Equation , 2010, MICCAI.

[44]  P. V. Yuldashev,et al.  Simulation of three-dimensional nonlinear fields of ultrasound therapeutic arrays , 2011, Acoustical physics.