1. Radially Polarized Spherical Piezoelectric Acoustic Transducer

2. This tutorial provides a step-by-step instruction to setup a 3D structural-acoustic interaction problem. Interaction between a vibrating piezoelectric structure with the surrounding fluid media is considered. Instructions on how to create a radially polarized piezoelectric material in spherical coordinates is provided. The coupled multiphysics is solved as a stationary problem by considering frequency domain analysis. Introduction

3. Schematic of a spherical transducer Piezoelectric element Surrounding fluid Ground Applied electric potential

4. Start from the Model Wizard

5. Number of waves we can capture We will use this information to create the modeling geometry such that we can capture two stationary waves in our model. Note that the larger the geometry, the higher the computation time and memory requirements.

6. Definitions These parameters will help us to define geometry, mesh and study settings

7. Geometry (1/4)

8. Geometry (2/4)

9. Geometry (3/4)

10. Geometry (4/4)

11. Materials Domains 2 and 3 Domain 1

12. Perfectly Matched Layer The outer layer of air domain is set as a PML. PML models an infinitely extended region that absorbs outgoing waves.

13. Spherical coordinates In order to model radial polarization of the piezo sphere, we need to define a spherical (local) coordinate system. The spherical coordinate directions will correspond to the local coordinates in the following manner. Local axisSpherical coordinates x1θ (Polar) x2φ (Azimuthal) x3r (Radial)

14. Creating a user-defined coordinate Local axis Spherical coordinates x1r (Radial) x2 θ (Polar) x3φ (Azimuthal) Model 1 > Definitions > Coordinate Systems > Base Vector System We will choose this one This will not give us the desired coordinate system in this case

15. Transformation functions Base vectorx,y,z components x1cos(atan2(y,x))*cos(acos(z/sqrt(x^2+y^2+z^2)))x-component sin(atan2(y,x))*cos(acos(z/sqrt(x^2+y^2+z^2)))y-component -sin(acos(z/sqrt(x^2+y^2+z^2)))z-component x2-sin(atan2(y,x))x-component cos(atan2(y,x))y-component 0z-component x3cos(atan2(y,x))*sin(acos(z/sqrt(x^2+y^2+z^2)))x-component sin(atan2(y,x))*sin(acos(z/sqrt(x^2+y^2+z^2)))y-component cos(acos(z/sqrt(x^2+y^2+z^2)))z-component We use the fact: φ = atan2(y,x) θ = acos(z/sqrt(x^2+y^2+z^2))

16. Setting up the coordinate system

17. Setting up the physics Choose Base Vector System 2 (sys2) from the Coordinate system drop-down menu

18. What happens to the material properties? Lets look at the coupling matrix e ij e 33 denotes the polarization along the x3-direction In COMSOLs global rectangular coordinate this would correspond to the global z-direction In the newly defined spherical coordinate this would correspond to the radial direction

19. Boundary condition - Ground Inner surface of piezo is grounded

20. Boundary condition – Electric Potential Outer surface of piezo is at a fixed potential

21. Boundary condition – Symmetry The symmetry planes take care of the fact that we are modeling only 1/8 th of the actual structure. This boundary condition only addresses the symmetry for the structural part of the model. We will look at relevant boundary conditions later that take care of the symmetry in the electrical and acoustic parts of the model.

22. Default boundary condition – Sound Hard Sound Hard Boundary (Wall) Reflecting boundary Assigned to outer surface of PML Also acts as the symmetry boundary for acoustics

23. Default boundary condition – Free Free Structurally free to deform Assigned to inner surface of piezo

24. Default boundary condition – Zero Charge Zero Charge No surface charge Assigned to symmetry faces of piezo Also acts as the symmetry boundary for the electrical part

25. Default boundary condition – Acoustic- Structure Boundary Acoustic-Structure Boundary This is automatically applied to the interface of the piezo (solid) and air (fluid) domains This boundary condition satisfies the following at the interface: -The normal force per unit area acting on the solid is equal to the normal fluid pressure -The normal acceleration of the fluid is equal to the normal acceleration of the solid

26. Mesh settings (1/2) This maximum element size would ensure that at least 5 elements are being used to account for each stationary wave.

27. Mesh settings (2/2)

28. Specify the frequency and compute You can solve this model for other frequencies as well since f0 has been defined as a parameter. The geometry and mesh will appropriately change to accommodate two stationary waves for the given frequency. The same idea holds true if you want to solve this model for any fluid other than air. For that you would need to change the value of the parameter c_fluid.

29. Results: Structural deformation of piezo The deformation shown here has been scaled by a factor of ~3000 for visualization purpose

30. Results: Electric potential on piezo

31. Results: Acoustic pressure

32. Results: Spherical coordinate system This shows that the coordinate system that we created represents a spherical coordinate system whose x3 direction (blue arrows) points along the radial direction. This allowed us to model radial polarization in a spherical piezo material.

33. Results: 1D pressure plot Pressure along the radial direction No discontinuity in pressure between the air domain adjacent to piezo and the PML Air domainPML