Premium
A multi‐time stepping integration method for the ultrasound heating problem
Author(s) -
Shevchenko I.,
Kaltenbacher M.,
Wohlmuth B.
Publication year - 2012
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201200023
Subject(s) - lagrange multiplier , robustness (evolution) , bioheat transfer , nonlinear system , bounded function , thermal conduction , multiplier (economics) , ultrasonic sensor , boundary value problem , heat equation , computer science , acoustic wave equation , mathematics , algorithm , acoustic wave , mathematical optimization , heat transfer , mathematical analysis , acoustics , physics , mechanics , biochemistry , chemistry , macroeconomics , quantum mechanics , economics , gene , thermodynamics
A two‐dimensional thermo‐acoustic problem consisting of the Pennes bioheat transfer model and a nonlinear wave equation with a temperature dependent speed of sound is considered. The large discrepancy in time scales of the wave propagation and the heat conduction processes requires a multi‐time stepping method allowing to solve both equations on different time scales. However, the standard approach considers a sequential solution process without any error control. We propose a multi‐time stepping scheme based on a fixed‐point approach resulting in a thermo‐acoustic coupling controlled by a stopping criterion. Furthermore, in most high‐intensity focused ultrasound applications the problem has to deal with absorbing boundary conditions (ABCs) in a bounded region. The use of a Lagrange multiplier based technique allows us to efficiently incorporate the second order Engquist–Majda ABC into the weak formulation of the original problem. The efficiency and robustness of the proposed multi‐time stepping method as well as improved accuracy compared to the widely used standard scheme for modeling high‐intensity focused ultrasound is demonstrated through a series of numerical examples which are typical for ultrasound heating.