(bultrasound)
This research project, supported by the National Agency for Research (AEI), Ministry of Science and Innovation of Spain (Agencia Estatal de Investigación, Ministerio de Ciencia e Innovación) and the European Regional Development Fund (FEDER) [grant number DPI2017-84758-P] , is a collaboration between several institutions and research groups worldwide. It is leaded by the Nanla research group (Nonlinear Analysis in Nonlinear Acoustics) at the Universidad Rey Juan Carlos in Spain, with specific tasks with the Université d'Artois (UA, FSA, LGCgE group) in France (numerical modeling tasks), the Centre National de la Recherche Scientifique (ISEN-IEMN-UMR-CNRS#9929, Acoustics lab) in France (numerical modeling tasks), and the Los Alamos National Laboratory (Acoustics & Sensors team) in the USA (experimental tasks). Several objectives are included in this research project, which spans from 01/2018 up to 09/2022.
The objective of this project is to add knowledge about the physical aspects related to the nonlinear interaction of ultrasound and gas bubbles in liquids, their effects and potential implications in medicine and industry, mainly through the developement of mathematical models for the simulation by numerical approximations. Concepts like parametric antennas, stable cavitation, characterization and detection of bubbly media, new mathematical models for the bubble dynamics have been studied with success. The most important achievements are: i) the development of specific numerical tools for the study of stable cavitation and Bjerknes forces in an three-dimensional ultrasonic field (including the nonlinear frequency mixing), of the generation of subharmonics in cavities (including their hysteretic character and optimization), of the nonlinear frequency mixing of three-dimensional focused ultrasonic fields, of the effects of a bubble cloud on a HIFU field, of the nonlinear resonance in the framework of the nonlinear mixing, of the nonlinear optimization of the medium; ii) the derivation and solution of a new differential equation for the bubble dynamics written in terms of volume variation, and its coupling with the wave equation for its nonlinear interaction with ultrasound in multibubbles populations; iii) the development of new methods for the solution of differential and algebraic nonlinear equations. The transfert of the results of this project is being carried out by means of collaborations with R&D companies and the application for patenting the new methods developed here. The JCR-SCI publications derived from the activities of our small research group during this project, to date, indicate 6 papers T1 and 1 T2 (4 Q1, 2 Q2, 1 Q3); 2 in the #1 position of the ranking. They add up to 31.569 in terms of Impact Factor. Othe publications are under preparation. Although this project has strongly been affected by the Covid-19 pandemic, The initial objectives and the new tasks that came up during its course have been achieved up to a great extend and they have been diffused through publications, congresses, visits, and webpage. Moreover, this project has generated new international collaborations, in the most theoretical aspects of our work as towards its potential applications, with other research institutions and R&D companies. It must be noted: (a) our current participation in an european application “MSCA Doctoral Networks 2022 (Horizon)” besides other european groups; (b) the large range of tasks open for the future open from the work carried out and the collaborations generated during this project. Some of them will be part of our new grant application to the National R&D Plan PID2022.
Objective#1. Nonlinear ultrasonic fields in bubbly liquids: parametric emission, stable cavitation, Bjerknes forces, self-focusing.
This objective includes the following tasks.
- The numerical analysis of the nonlinear mixing of focused ultrasonic signals of different frequencies in three-dimensional bubbly liquids from a spherical source [Paper#1]. The evolution beyond the focus of both the difference and sum frequency components, not present at the dual-ultrasonic frequency source, is the main goal of this study, especially in relation to attenuation (Fig. 1) (Paper#1). The new signal takes advantage of the combination of each kind of components, the directivity of the high-frequency ones generated nonlinearly and the penetrability of the low-frequency ones.
- The three-dimensional numerical study of the effects of a bubble cloud on a high intensity focused ultrasonic field (HIFU), such as the modification of the size and position of the focal region, through the evaluation of the mechanical index (Fig. 2), which estimates the dammage caused in tissues (Paper#3).
- The numerical study of the nonlinear resonance effect of a standing wave in a resonator filled with a bubbly liquid, especially in relation to the difference and sum frequency components from a dual-frequency source, which are affected by the shift of resonance when amplitudes are raised (Fig. 3), characteristic of the softening of the medium. This effect allows the amplification and maximization of the signals at these frequencies (Paper#2, Paper#5) (Figs. 4 y 5).
- The study of numerical methods for solving nonlinear equations (Paper#4, Paper#7).
- The numerical analysis of three-dimensional focused ultrasound of finite amplitude in bubbly liquids through a finite-element model (Fig. 6) (Paper#12). This work will allow us to model different bubble densities and sizes thouroughly, and to consider more realistic cavities such as the ones used for R&D in sonochemistry, which will be part of our new grant application to the National R&D Plan PID2022. This model will be useful for the study of the directivity of signals generated nonlinearly in a bubbly liquid by modifications of the boundary conditions in the numerical models or by variations of the bubble density near the source and for the study of the self-focusing from a sonotrode of low-ultrasonic frequency, by considering different bubbly structures at the source (layers and other geometries observed experimentally)
- The numerical study on the generation of subharmonics in a cavity filled with a bubbly liquid (collaboration with Centre Rapsodee, CNRS, France) (Figs. 7 & 8), considering a single-frequency source in reflecting-walled cavities, which shows the threshold dependence and hysteretic behavior of the generation of subharmonic components vs. pressure amplitude (Paper#11). Further studies in this framework will be part of our new grant application to the National R&D Plan PID2022.
- The development of a three-dimensional numerical model for the analysis of the stable cavitation process in a focused ultrasonic field. Our results show the creation of bubble clouds (Fig. 10) and the effect of the primary Bjerknes forces of these bubbles (Fig. 11) (Paper#6). See also Vid. 1 and 2. This work allows us to plan new tasks that will be part of our new grant application to the National R&D Plan PID2022.
- The development of a numerical model for the simulation of the stable cavitation process by taking into account the nonlinear mixing of several signals of different frequencies, assuming a focused ultrasonic field or a resonator, which is still under developement.
- The numerical study of the propagation of solitary waves in multi-dimensional volumes of bubbly liquids (Fig. 12), which is still under development.
Objective#2. Characterization and detection of bubbles in ultrasonic fields.
Regarding the detection of bubble clusters, the conclusions of the analysis carried out on three-dimensional focused ultrasound of different frequencies with our numerical models, and their effects on the attenuation of the waves beyond the focus, mainly the difference and sum components, and the analysis carried out by varying the parameters of the problems, especially the spherical source, with a single and dual-frequency source, the source amplitude (Paper#1), open a way for the detection of bubbles which shows the presence of a high nonlinear medium due to the gas bubbles. This kind of studies will be part of our new grant application to the National R&D Plan PID2022. At the same time, the modification of the size and location of the focus obtained by HIFU due to the presence of a bubble cloud could be used as a detection method, taking into account that these bubbles are generated by cavitation in the high-intensity ultrasonic field (Paper#3).
A study about the nonlinear characterization of bubbly liquids, for their optimization, through the use of our numerical models has been carried out. The results obtained allows us to define the optimum medium, in terms of bubble size and density, to maximize the harmonics generated during the propagation of finite-amplitude ultrasound in several configurations (free-walled cavity and open-field case) (Fig. 14) (Paper#10, Paper#12). The continuity of these studies about subharmonics (*) and parametric antennas will be part of our new grant application to the National R&D Plan PID2022.
(*) The numerical study on the optimization (amplification) of subharmonics, by taking advantage of the softening of the medium when amplitudes are raised has been carried out by frequency sweeping in a resonant cavity (Fig. 9) (Paper#13). Multidimensional cavities with other boundary conditions are of interest and they will most likely be considered in future studies.
This objective also considers two kinds of studies for the characterization of bubbly liquids that are still under development. The unstable bubbles in the liquid induces many experimental issues to obtain repetitive measurements. For a single isolated gas bubble, the analysis of the primary Bjerknes forces due to an ultrasonic field in a liquid (radiation forces) allows to observe the bubble deflection. The experimental setup built for this study includes a tank, a system that produces gas bubbles, one by one from the bottom of the tank, and an ultrasonic source on one side of the tank. The system is tested with air bubbles in mineral oil with a continuous sine driving signal at 179.1 kHz, i.e., the third radial mode, which induces a high directivity. The next step for this task is the adaptation of our numerical codes to the real conditions of the experiment. For a population of gas bubbles, the aim is the nonlinear characterization of a bubbly liquid made of Fluorinert FC-43 and Helio gas bubbles. Our goal is to compare to the numerical results already obtained. Two different experimental setup have been considered, both including a tank, a system based on porous metal that produces a dense population of bubbles from the bottom of the tank, and an ultrasonic source placed on one side of the tank in front of a needle hydrophone. The system is tested with a sine burst driving signal at ~1 MHz. The "slope" method is used to determine the nonlinear parameter of the medium.
Objective#3. Higher-order approximations for the bubble dynamics.
A new model up to the fourth-order approximation for the differential equation defining the behavior of a gas bubble, written in terms of bubble volume variation, has been developed. The goal here is the derivation of a Rayleigh-Plesset equation, which approximation is of order higher than the existing ones. This work, including the derivation of the equation and its numerical solution, has been published (Paper#9). The new equation shows differences with the model of lower approximation order (the one generally used in works carried out with bubble volume) when the equation is solved through numerical approximations, which reveals the conditions for which the new model is useful and necessary. These differences show up when the nonlinearity is more important, i.e., when the driving frequency is close to the bubble resonance. Fig. 15 shows the response of the bubble obtained with the new model to the excitation of the acoustic field as the amplitude of the source increases, and its differences around the bubble resonance compared to an existing model.
These Rayleigh-Plesset equations of different order of approximation written in terms of bubble volume variation have been solved through the development of a linear iterative method, which allows us to observe (Figs. 16 and 17) and compare the advantages of the solutions obtained with each model. The first case, corresponding to the second-order approximation, has been published (Paper#8), whereas the second one, corresponding to the fourth-order approximation, is still under development.
The incorporation of the new equation of fourth-order approximation into the numerical models developed by the group, by coupling it with the wave equation, for the study of the nonlinear interaction of a population of bubbles and an ultrasonic field, has been performed. The model is under testing and the physical interpretation of the benefits obtained from the new approximations, which are expected to be important close to the bubble resonance, will be studied in the near future. They will most likely be part of our new grant application to the National R&D Plan PID2022.
The results obtained have been published in
Papers:
#1) M.T. Tejedor Sastre and C. Vanhille, Numerical study of frequency mixing in a focused ultrasonic field in bubbly liquids from a dual-frequency spherical source, Results in Physics 11, 726-733 (2018) (Impact Factor JCR: 3.042; T1 Q1)
#2) M.T. Tejedor Sastre and C. Vanhille, Nonlinear resonance of cavities filled with bubbly liquids: A numerical study with application to the enhancement of the frequency mixing effect, Shock and Vibration 2018, 1570508 (2018) (Impact Factor JCR: 1.628; T2 Q3)
#3) C. Vanhille and K. Hynynen, Numerical simulations of the nonlinear interaction of a bubble cloud and a high intensity focused ultrasound field, Acoustics 2, 825-836 (2019) (in JCR, but with no Impact Factor yet)
#4) C. Vanhille, A note on the convergence of the irrational Halley’s iterative algorithm for solving nonlinear equations, International Journal of Computer Mathematics 97, 1840-1848 (2020) (Impact Factor JCR: 1.931; T1 Q2)
#5) M.T. Tejedor Sastre and C. Vanhille, Nonlinear maximization of the sum-frequency component from two ultrasonic signals in a bubbly liquid, Sensors 20, 113 (2020) (Impact Factor JCR: 3.576; T1 Q1)
#6) C. Vanhille, Numerical simulations of stable cavitation bubble generation and primary Bjerknes forces in a three-dimensional nonlinear phased array focused ultrasound field, Ultrasonics Sonochemistry 63, 104972 (2020) (Impact Factor JCR: 7.491; T1 Q1 (#1))
#7) C. Vanhille, An improved secant algorithm of variable order to solve nonlinear equations based on the disassociation of numerical approximations and iterative progression, Journal of Mathematics Research 12, 50-65 (2020) (not in JCR yet, but peer-reviewed journal)
#8) C. Vanhille, Linear iterative procedure to solve a Rayleigh-Plesset equation, Acoustics 3, 212-220, 2021 (in JCR, but with no Impact Factor yet)
#9) C. Vanhille, A fourth-order approximation Rayleigh-Plesset equation written in volume variation for an adiabatic-gas bubble in an ultrasonic field: Derivation and numerical solution", Results in Physics 25, 104193 (2021) (Impact Factor JCR: 4.565; T1 Q2)
#10) M.T. Tejedor Sastre and C. Vanhille, An analysis of nonlinear ultrasonic waves in bubbly liquids by a numerical model, Revista de Matemática: Teoría y Aplicaciones 29, en prensa (2022) (not in JCR, but peer-reviewed journal)
#11) M.T. Tejedor Sastre, O. Louisnard, and C. Vanhille, Generation of subharmonics in acoustic resonators containing bubbly liquids: A numerical study of the excitation threshold and hysteretic behavior, Ultrasonics Sonochemistry 88, 106068 (2022) (JCR#1 Q1-T1, IF=9,336)
#12) M.T. Tejedor Sastre, A. Leblanc, A. Lavie, and C. Vanhille, Modeling and simulation of parametric nonlinear focused ultrasound in three-dimensional bubbly liquids
with axial symmetry by a finite-element model, in drafting
#13) M.T. Tejedor Sastre and C. Vanhille, Definition of bubbly liquids parameters for the optimization of their nonlinear effects on ultrasound, in drafting
Sum of Impact Factor JCR: 31.569
and Conference Proceedings:
#i) M.T. Tejedor Sastre, O. Louisnard, and C. Vanhille, Une étude numérique de la génération de sous-harmoniques en liquides bulleux au sein de résonateurs ultrasonores, Actes des Journées Scientifiques Ultrasons et Procédés – 5ième édition, 2019
#ii) M.T. Tejedor Sastre and C. Vanhille, A numerical study of the nonlinear behavior of ultrasound in bubbly liquids, Proceedings of the Congress on Numerical Methods in Engineering, 2019
#iii) M.T. Tejedor Sastre and C. Vanhille, Numerical models for nonlinear ultrasound in bubbly liquids, Proceedings of the International Congress on Industrial and Applied Mathematics, 2019
#iv) C. Vanhille and M.T. Tejedor Sastre, Numerical modelling for the simulation of nonlinear ultrasound in liquids with gas bubbles, Proceedings of the 2nd International Conference on Numerical Modelling in Engineering, IOP Conference Series: Materials Science and Engineering 657, 012006, 2019
#v) M.T. Tejedor Sastre and C. Vanhille, An analysis of nonlinear ultrasonic waves in bubbly liquids by a numerical model, Proceedings of the XXIII International Symposium on Mathematical Methods Applied to the Sciences, 2022
#vi) M.T. Tejedor Sastre and C. Vanhille, Amplitude-dependent effects of ultrasound in bubbly liquids, Proceedings of the 30th Symposium on Vibrations in Physical Systems, 2022
María Teresa Tejedor Sastre earned her Ph.D. "Modelos numéricos para el estudio de ondas ultrasónicas no lineales en líquidos con burbujas: Aplicación a la mezcla de frecuencias" (Numerical models for the study of nonlinear ultrasonic waves in bubbly liquids: Application to frequency mixing) on 01/23/2019, Universidad Rey Juan Carlos, Escuela Internacional de Doctorado ("Sobresaliente Cum Laude por unanimidad"; Mención Internacional; Premio Extraordinario de Doctorado del Programa de Doctorado en Ciencias) (supervisor: Christian Vanhille).
Figures:
vidp.avividN.avi