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Numerical Simulation of Acoustics-Acoustics- and Structural Mechanics-Acoustics-Couplings on Nonmatching Grids. Part I.

Project manager: Prof. Dr. rer. nat. B. I. Wohlmuth
Projektbearbeiter: Dr. B. Flemisch, Dr. I. Shevchenko
Projektpartner: Lehtstuhl für Sensorik, Universität Erlangen-Nürnberg

Project description

The project is aimed at investigation and implementation of new numerical methods for efficient solution of the acoustic- and mechanical-acoustic coupled models on nonconforming grids.

Current results

The numerical simulations of acoustic- and mechanical-acoustic coupled models usually confront with spatial-discretization problems due to various space-scales in different regions of the computational domain. It may take place, for instance, when computing a wave propagation generated by a transmitter. The spatial region where the wave is propagated can be discretized with a coarse grid while the computation domain of the transmitter must be performed using a much fine grid. A demonstrative example of such different space-scales regions is the emission of acoustic waves by multiple structures which admits the steering of the waves by exciting the structures in a specified chronological order (figure 1).

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Figure 1. Cylindrical plates attached to the fluid domain (left), isosurfaces of the acoustic potential, deformed plates (right)

If one had to employ matching grids, it would be quite difficult to generate them, and if the mesh-width could not be very small over the whole domain, the resulting element shapes would possibly result in a poor approximation of the solution. The nonconforming approach admits to use the grid desired for each subdomain regardless of the grids for the other subdomains. Moreover, it is very easy to add more plates or to change their position. Only the corresponding part of the coupling matrix would have to be (re-)calculated. Figures 2 and 3 show snapshots, taken every 10 time steps of 3.5 ns, of the evolution of the acoustic velocity potential along with the deformation of the structures (magnified by a factor of 1000).

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Figure 2. Evolution of the acoustic velocity potential and of the deformed structures, synchronous excitation: snapshots after 10, 20, ... ,80 timesteps.

For the results presented in Figure 2, the cylindrical plates are excited simultaneously, while for Figure 3, they are excited successively.

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Figure 3. Evolution of the acoustic velocity potential and of the deformed structures, successive excitation: snapshots after 10, 20, ... ,80 timesteps.

For both calculations, the waves emitting from the structures add up as expected to constitute the superposed global sound beam. Given a target point, it is possible to optimally steer the acoustic wave towards this point by appropriately adjusting the chronological order of excitation of the silicon chips. This principle is used in so-called capacitive micro-machined ultrasound transducers. There, the deformation of the structure is induced by an electrostatic surface force acting on the boundary.

Numerical Simulation of Acoustics-Acoustics- and Structural Mechanics-Acoustics-Couplings on Nonmatching Grids. Part II.

Project manager: Prof. Dr. rer. nat. B. I. Wohlmuth
Projektbearbeiter: Dr. I. Shevchenko
Projektpartner: Applied Mechatronics, Alps-Adriatic University of Klagenfurt

Project description

The project is aimed at investigation and implementation of new numerical methods for efficient solution of the acoustic- and mechanical-acoustic coupled models on nonconforming grids.

Current results

Multi-time stepping integration in multiphysics problems. The heating of media by high-intensity focused ultrasound waves offers a desirable effect for a variety of applications. For instance, the ultrasonic thermotherapy of cancer in which destruction of soft tissues in the body is effected through high temperatures generated by the local absorption of wave energy. Within the framework of the project, we investigate the Pennes bio-heat transfer equation coupled with the non-linear Kuznetsov wave equation. In this multiphysics problem, an ultrasound transducer, located on the bottom of the computational domain (see Fig.4), generates waves of frequency 1 MHz which are propagated from bottom to top and heat up the tissue. Figure 4 shows snapshots, taken every 1.0 ms, of the evolution of the temperature and acoustic fields in the liquid (lower subdomain) and in the tissue (upper subdomain).

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Figure 4. A series of snapshots after 1.0 ms, 2.0 ms, 3.0 ms, 4.0 ms and 5.0 ms of insonation. The evolution of the temperature and acoustic fields.

Wave propagation and heat conduction are the processes which occur on different time scales. The temperature field evolves quite slowly and can be fully resolved in the range of seconds. On the contrary, the ultrasound waves are emitted at frequencies from 100 kHz to 10 MHz. That makes the time scales incomparable and requires special attention being paid to numerical methods for such kind of problems. Within the framework of the project, a multi-time stepping integration method was developed. It has been demonstrated that the proposed multi-time stepping method is more accurate compared to the conventional non-iterative technique using one time step to advance acoustic and temperature fields. Moreover, it has been also shown that the new method has lower computational complexity compared to the full Newton scheme and works faster for large ratios the time scales compared to the standard approach. We would also like to stress out that the proposed method can be used not only for the ultrasound heating problem but also in other multiphysics models which exhibit processes evolving on substantially different time scales.

Self-adapting absorbing boundary conditions for wave equations. Within the framework of the project we proposed a self-adapting absorbing boundary condition (ABC) for the linear wave equation. The construction of ABC is based on a local computation of the incidence angle of the outgoing wave and on the use of the classical lowest order Engquist--Majda absorbing boundary condition. In order to obtain a good approximation of the incidence angle, we decompose adaptively the absorbing boundary into subsegments and locally apply the Fourier transformation.

In Fig.5, a monofrequency transducer is located on the left vertical boundary of the computational domain which emits sinusoidal acoustic waves traveling from the left to the right. The absorbing boundary, inclined at the angle 30 degree to the horizontal axis, is set on the right boundary of the domain. Rows a and b stand for the first and second order Engquist--Majda ABCs while in row c we show the results of the newly designed self-adapting ABC. For this setup, the reference solution is a plane wave.

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Figure 5. The evolution of the acoustic potential for the absorbing boundary inclined at 30 degree to the horizontal axis. The time t is in microseconds

The developed self-adapting ABC is attractive from the computational point of view due to its local character and is easy to implement. Due to its flexibility it can handle more complex situations and provides quantitatively much better results compared to the classical first and second order Engquist--Majda ABCs. In the case of non-trivial geometries, the decomposition of the boundary into segments improves considerably the performance of the developed ABC. The principal feature of this method is that it can be applied to various types of ABCs which can be build for a specific angle of incidence. For instance, this approach can be used with ABCs for wave equations with variable coefficients derived through the pseudo-differential approach proposed by Engquist and Majda in 1977. Furthermore, the developed method is applicable to ABCs used for nonlinear wave equations the derivation of which can also be based on the pseudo-differential calculus.

Future research

Further research will be concentrated on the development of a coupled thermo-acoustic-elastic problem and numerical methods for its efficient treatment.