1. Plane Wave Echo Particle Image Velocimetry Samuel Rodriguez, Xavier Jacob, Vincent Gibiat PHASE University Paul Sabatier

2. Basics of topological optimisation applied to acoustic waves Topological optimisation: optimisation of a physical domain for a given set of loads and boundaries Numerical applications for electromagnetic and ultrasonic imaging [Pommier and Samet, Bonnet, Malcolm and Guzina, Dominguez and Gibiat, Sahuguet Chouippe and Gibiat] An experimental application with a transducer array: the TDTE method [Dominguez and Gibiat, Dominguez Gibiat and Esquerre]. Use of a time- domain finite-difference model. The Fast Topological Imaging method is an adaptation in the frequency domain of the TDTE method that aims at reducing the computation cost. 2 S. Rodriguez, X. Jacob, V. Gibiat Plane Wave Echo Particle Image Velocimetry Basics of topological optimization applied to acoustic waves

3. 3S. Rodriguez, X. Jacob, V. Gibiat Plane Wave Echo Particle Image Velocimetry Topological optimization Initial domain Parameterization Shape optimization Topological optimization Figure adapted from [J. Pommier, “L’asymptotique topologique en electromagnétisme”, PhD thesis] Basics of topological optimization applied to acoustic waves

4. 4S. Rodriguez, X. Jacob, V. Gibiat Plane Wave Echo Particle Image Velocimetry Topological optimization Figure adapted from [J. Pommier, “L’asymptotique topologique en electromagnétisme”, PhD thesis] Solution without a “hole” Solution with a “hole” Cost Calculation of the gradient Basics of topological optimization applied to acoustic waves

5. 5S. Rodriguez, X. Jacob, V. Gibiat Plane Wave Echo Particle Image Velocimetry Referenc ee 00 u (,t) Inspected Med. ? mm u m (,t) 2- Numerical computation of the reference field and measure ofu(r,t) Adjoint Prob. (u m -u )(,t) (u m -u )(,T-t) 00 3- Difference between ref and inspected then time reversal to compute Adjoint v (,t) Calcul of topological derivative in time domain 4- the adjoint field v(r,t) 1- Echographic measure of u m (r,t) Basics of topological optimization applied to acoustic waves

6. How does it work in “true life” Experimental conditions –32-transducer array. Resonance freq 5 MHz. 0.8 mm pitch. –Lecoeur OPEN system 80 MHz. –Plane wave. 3-period sinus. 6 Transducer array Time Gelatin cylinder Array Water Plane Wave Echo Particle Image Velocimetry Experimental static results

7. 7S. Rodriguez, X. Jacob, V. Gibiat Plane Wave Echo Particle Image Velocimetry How to take into account the geometry and the radiation of the transducers? How to compute efficiently (fast and accurate) the direct and adjoint fields ? A solution is to transpose the time domain to the frequency one TDTE versus FTIM Experimental static results

8. we have the physical information that comes from : –The experimental data: –Dimensions of the transducers and a theoretical or a numerical model (as near as possible from the reality) of the wave propagation in the medium 1 ) Computation of the radiation patterns of every transducer j and every frequency : 8 FT signal emitted by transducer j FT signal measured with transducer j Transducer COMPUTED ONCE AND FOR ALL Plane Wave Echo Particle Image Velocimetry Experimental static results

9. 2.Computation of the solutions with simple multiplications (  time-domain convolutions) : 9 X X X + + Transducer array Plane Wave Echo Particle Image Velocimetry Experimental static results

10. 3.Computation of the topological derivative of the FTIM method 10 Time Depth Transducer array Envelope of RF signalsFTIM Plane Wave Echo Particle Image Velocimetry Experimental static results

11. Application to an anisotropic solid medium Composite material sample Radiation patterns computed with a FE model. 11 TDTEFTIM² 100 TIMES FASTER Plane Wave Echo Particle Image Velocimetry Experimental static results

12. 12S. Rodriguez, X. Jacob, V. Gibiat Plane Wave Echo Particle Image Velocimetry Experimental dynamic results Small water tank Put marble powder « beatite from Saint Béat » Let the bigger particles sediment Particles smaller than 40 micrometers (invisible) remain in water Insonification from the bottom Image of a slice of the water tank

13. 13S. Rodriguez, X. Jacob, V. Gibiat Plane Wave Echo Particle Image Velocimetry Experimental dynamic results Sedimentation of marble powder Water level Bottom Top

14. 14S. Rodriguez, X. Jacob, V. Gibiat Plane Wave Echo Particle Image Velocimetry Experimental dynamic results Passage of a single wave at the water surface The interface water/air acts as a mirror Water level Top

15. 15S. Rodriguez, X. Jacob, V. Gibiat Plane Wave Echo Particle Image Velocimetry Experimental dynamic results Water rotated with a magnetic agitator and seeded with small particles (about 40 micrometer big), mimicking contrast agents. PRF=250 images/s, and horizontal insonification video_vortex_flow

16. 16S. Rodriguez, X. Jacob, V. Gibiat Plane Wave Echo Particle Image Velocimetry Conclusion Instead of Time Domain Topological Energy (10 minutes/image) Frequency Domain alternative is possible (FTIM) (6 seconds/image) Through FTIM algorithm it is possible to record sequences at frequency varying between 250 Hz and 1000 kHz to derive dynamic ultrasonic images of moving very small particles Everywhere such “reflecting” objects exist it is possible to image Their movements FTIM is a credible alternative to PIV each time it is not possible to optically Illuminate the medium