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Ultrasonic Flowmeter with Piezoelectric Transducers: Coupling Between FEM and DG
Introduction
Ultrasonic flowmeters are used to determine the velocity of the fluid flowing through a pipe. The principle is to send an ultrasonic signal across the flow at a skew angle. In case of no flow, the transmitting time between the transmitter and the receiver is the same for the signals sent in the upstream and the downstream directions. Otherwise, the downstream traveling wave moves faster than the one traveling upstream. In many cases piezoelectric transducers are used to send and receive the ultrasonic wave.
This tutorial example shows how to simulate an ultrasonic flowmeter with piezoelectric transducers in the simplified no-flow case. The finite element method (FEM) is used to model the piezoelectric transducers, whereas the modeling of the ultrasonic wave propagation is based on the discontinuous Galerkin (DG) method. The whole model is split into three submodels. A FEM to DG coupling is used to send the wave from the transmitter, and a DG to FEM coupling is used for the receiver.
Model Definition
The flowmeter geometry used in this model is the same as given in the model Ultrasound Flowmeter with Generic Time-of-Flight Configuration. It consists of a main pipe and a signal pipe of a smaller diameter. The signal tube is tilted to the main pipe at the angle α = 45o. Two piezoelectric transducers placed at either end of the signal pipe operate as the transmitter and the receiver. An input voltage signal applied to the transmitter results in the mechanical deformation of the piezoelectric transducer, which is due to the inverse piezoelectric effect. The mechanical deformation in its turn generates an acoustic wave in the fluid. When the acoustic wave reaches the receiver, the inverse process takes place: the mechanical load is being converted into an electric signal because of the direct piezoelectric effect.
As previously mentioned, this model studies the propagation of the acoustic wave is the simplified no-flow condition. Thus the main focus is the interaction between the fluid and the solid and the conversion input electric signal - acoustic wave - output electric signal. The background flow can be added if necessary, as is done in the model Ultrasound Flowmeter with Generic Time-of-Flight Configuration.
This model solves the full three-dimensional problem with the symmetry condition imposed on the sagittal plane of the structure. The model involves the following physical phenomena.
Piezoelectricity: solid mechanics and electrostatics
The transmitter and the receiver can be used interchangeably; therefore, the principles of their construction are the same. The main part is a disk made of a piezoelectric material (for example, PZT) that is used for conversion of between the electric and the mechanical waves.
A direct propagation of the mechanical wave in the piezoelectric transducer in form of the acoustic wave in the fluid will result in significant reflections from the solid/fluid interface and, consequently, losses. This is due to the impedance mismatch: for water, the characteristic acoustic impedance  kg/m2·s, and for PZT-5H,  kg/m2·s. Therefore, a matching layer is required to minimize the losses. Its size is taken to be 1/4 of the wavelength, and its impedance is calculated as follows Ref. 1:
(1).
The impedance matching itself is a problem that requires a though investigation and is out of scope of this model. Here, the acrylic plastic is used as the matching layer material.
The input signal applied to the transmitter is a harmonic voltage pulse of the amplitude V0 = 50 V, the frequency f0 = 2.5 MHz, and duration of 2 μs. The voltage profile is depicted in Figure 1.
acoustics: convected wave equation, time explicit
The vibrations of the transmitter generate an ultrasonic acoustic wave traveling from the transmitter to the receiver. The propagation of the acoustic wave is defined by the normal velocity boundary condition prescribed on the interface between the matching layer of the transmitter and the fluid domain. The velocity components are derived from the submodel that simulates the transmitter.
When the acoustic signal reaches the other end of the signal tube, it will interact with the receiver. The acoustic pressure imposed on the interface between the fluid domain and the matching layer of the receiver will be used as the boundary condition in the receiver submodel.
Figure 1: Input voltage applied to the transmitter.
Results and Discussion
The propagation of the ultrasonic acoustic wave near the transmitter at time t = 2.4 μs is shown in Figure 2. The driving voltage applied to the transmitter is that depicted in Figure 1.
The propagation of the ultrasonic pressure wave over the symmetry plane of the flowmeter is depicted in Figure 3. Note that the upper left plot presents the full profile of the signal shown in Figure 2.
By the time t = 10 μs the pressure signal has reached the receiver. Figure 4 shows the mechanical deformation profile of the receiver. In Figure 5 you can see the profiles of the input voltage applied to the transducer and the output electric signal read on the receiver. As previously mentioned, the time of flight will be different for the signal propagating downstream and the one propagating upstream in the presence of a background flow.
Figure 2: Propagation of the ultrasonic acoustic wave from the transmitter.
Figure 3: Propagation of the acoustic pressure signal at 4 time steps.
Figure 4: Mechanical deformation of the receiver.
Figure 5: Input and output electric signals as functions of time.
Notes About the COMSOL Implementation
The described split of the entire model into three submodels has a certain reason. The parts that model the piezoelectric transducers use the conventional FEM, whereas the ultrasonic wave propagation is modeled by the DG method. The time-dependent solvers for both approaches are different, which makes the direct FEM to DG or DG to FEM couplings impossible.
The workaround discussed in this tutorial is based on the indirect coupling between the models that use different numerical schemes. This is done through the identity mapping coupling operators idmap1 and idmap2. These are used on the transmitter-fluid and the fluid-receiver interfaces, respectively. Thus the operators map the results between the submodels and provide the required couplings.
The following points of the submodels’ setups must be taken into account:
•
The transmitter submodel includes the domains for the piezoelectric transducer, the matching layer, and the fluid layer. The last is located on top of the matching layer and is used to account the attached fluid mass providing the fluid-structure interaction. The Pressure Acoustics, Transient interface solves for the pressure wave in the fluid. The fluid domain is truncated by a perfectly matched layer. Note that this approach remains appropriate even in the case of a background flow, because the considered fluid domain is occupies only a small part of the sensing tube adjacent to the transmitter. Therefore, the interaction between the fluid attached to the matching layer and the background flow in the main pipe is negligible.
•
In order to account for the interaction between the ultrasonic pressure wave and the receiver, an impedance boundary condition is imposed on the fluid-receiver interface in a Convected Wave Equation, Time Explicit interface.
Reference
1. V. A. Gavrilova, M. G. Fazlyyyakhmatov, and N. F. Kashapov, “Protective matching polymer powder coating of piezoelectric element,” J. Phys.: Conf. Ser., vol. 479, issue 1, 2013.
Application Library path: Acoustics_Module/Ultrasound/flow_meter_piezoelectric_transducers
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Blank Model.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file flow_meter_piezoelectric_transducers_parameters.txt.
Specify the driving voltage applied to the transmitter.
Rectangle 1 (rect1)
1
In the Home toolbar, click  Functions and choose Global>Rectangle.
2
In the Settings window for Rectangle, type rect_fct in the Function name text field.
3
Locate the Parameters section. In the Lower limit text field, type 0.5e-6.
4
In the Upper limit text field, type 1.5e-6.
5
Click to expand the Smoothing section. In the Size of transition zone text field, type 1e-6.
Analytic 1 (an1)
1
In the Home toolbar, click  Functions and choose Global>Analytic.
2
In the Settings window for Analytic, type voltage_pulse_fct in the Function name text field.
3
Locate the Definition section. In the Expression text field, type V0*sin(omega0*t)*rect_fct(t).
4
In the Arguments text field, type t.
5
Locate the Units section. In the Arguments text field, type s.
6
In the Function text field, type V.
7
Locate the Plot Parameters section. In the table, enter the following settings:
Argument
Lower limit
Upper limit
t
0
10*T0
8
Click  Plot.
The input electric signal should look like the one in Figure 1.
The entire model is split into submodels. First, create a model that simulates the propagation of the acoustic signal induced by the input signal applied to the transmitter.
Add Component
In the Home toolbar, click  Add Component and choose 3D.
Geometry 1
The geometry will contain the domains corresponding to the piezoelectric transducer, the matching layer, and a layer of the fluid in the signal tube. The last is used to account for the attached fluid mass and model the fluid-structure interaction.
Cylinder 1 (cyl1)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type D_transducer/2.
4
In the Height text field, type lam0+L_pml.
5
Locate the Position section. In the x text field, type L/2.
6
In the z text field, type -L_transducer/2.
7
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
L_pml
8
Clear the Layers on side check box.
9
Select the Layers on top check box.
Cylinder 2 (cyl2)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type D_transducer/2.
4
In the Height text field, type L_matching.
5
Locate the Position section. In the x text field, type L/2.
6
In the z text field, type -L_transducer/2-L_matching.
Cylinder 3 (cyl3)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type D_transducer/4.
4
In the Height text field, type L_piezo.
5
Locate the Position section. In the x text field, type L/2.
6
In the z text field, type -L_transducer/2-L_matching-L_piezo.
Rotate 1 (rot1)
1
In the Geometry toolbar, click  Transforms and choose Rotate.
2
In the Settings window for Rotate, locate the Rotation section.
3
In the Angle text field, type alpha.
4
Locate the Point on Axis of Rotation section. In the x text field, type L/2.
5
Locate the Rotation section. From the Axis type list, choose y-axis.
6
Click in the Graphics window and then press Ctrl+A to select all objects.
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
From the Plane list, choose zx-plane.
Partition Objects 1 (par1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Partition Objects.
2
Click in the Graphics window and then press Ctrl+A to select all objects.
3
In the Settings window for Partition Objects, locate the Partition Objects section.
4
From the Partition with list, choose Work plane.
Delete Entities 1 (del1)
1
In the Model Builder window, right-click Geometry 1 and choose Delete Entities.
2
In the Settings window for Delete Entities, locate the Entities or Objects to Delete section.
3
From the Geometric entity level list, choose Domain.
4
On the object par1(1), select Domains 1 and 3 only.
5
On the object par1(2), select Domain 1 only.
6
On the object par1(3), select Domain 1 only.
7
Click the  Go to YZ View button in the Graphics toolbar three times to view the geometry from below.
8
Click the  Go to Default View button in the Graphics toolbar.
9
In the Geometry toolbar, click  Build All.
10
Click the  Zoom Extents button in the Graphics toolbar.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in>Water, liquid.
4
Click Add to Component in the window toolbar.
5
In the tree, select Built-in>Lead Zirconate Titanate (PZT-5H).
6
Click Add to Component in the window toolbar.
7
In the tree, select Built-in>Acrylic plastic.
8
Click Add to Component in the window toolbar.
9
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
Lead Zirconate Titanate (PZT-5H) (mat2)
1
In the Model Builder window, under Component 1 (comp1)>Materials click Lead Zirconate Titanate (PZT-5H) (mat2).
2
Select Domain 3 only.
It might be easier to select the correct domain by using the Selection List window. To open this window, in the Home toolbar click Windows and choose Selection List. (If you are running the cross-platform desktop, you find Windows in the main menu.)
Acrylic plastic (mat3)
1
In the Model Builder window, click Acrylic plastic (mat3).
2
Select Domain 1 only.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select Acoustics>Pressure Acoustics>Pressure Acoustics, Transient (actd).
4
Click Add to Component 1 in the window toolbar.
5
In the tree, select Structural Mechanics>Solid Mechanics (solid).
6
Click Add to Component 1 in the window toolbar.
7
In the tree, select AC/DC>Electric Fields and Currents>Electrostatics (es).
8
Click Add to Component 1 in the window toolbar.
9
In the Home toolbar, click  Add Physics to close the Add Physics window.
Pressure Acoustics, Transient (actd)
1
In the Model Builder window, under Component 1 (comp1) click Pressure Acoustics, Transient (actd).
2
Select Domains 2 and 4 only.
Solid Mechanics (solid)
1
In the Model Builder window, under Component 1 (comp1) click Solid Mechanics (solid).
2
Select Domains 1 and 3 only.
Electrostatics (es)
1
In the Model Builder window, under Component 1 (comp1) click Electrostatics (es).
2
Select Domain 3 only.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies>Time Dependent.
4
Click Add Study in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Definitions
Transmitter
1
In the Model Builder window, expand the Component 1 (comp1)>Definitions node.
2
Right-click Definitions and choose Selections>Explicit.
3
In the Settings window for Explicit, locate the Input Entities section.
4
From the Geometric entity level list, choose Boundary.
5
Select Boundary 6 only.
6
In the Label text field, type Transmitter.
Solid
1
In the Definitions toolbar, click  Explicit.
2
Select Domains 1 and 3 only.
3
In the Settings window for Explicit, type Solid in the Label text field.
The fluid domain is truncated by introducing a perfectly matched layer (PML) which absorbs the propagating acoustic waves in the fluid.
Perfectly Matched Layer 1 (pml1)
1
In the Definitions toolbar, click  Perfectly Matched Layer.
2
Select Domain 4 only.
3
In the Settings window for Perfectly Matched Layer, locate the Scaling section.
4
From the Coordinate stretching type list, choose Rational.
Define a coordinate system that corresponds to the piezoelectric material orientation: the z-axis of the piezoelectric crystal points along the signal tube axis.
sys_PZT
1
In the Definitions toolbar, click  Coordinate Systems and choose Base Vector System.
2
In the Settings window for Base Vector System, type sys_PZT in the Label text field.
3
Locate the Settings section. Find the Simplifications subsection. Select the Assume orthonormal check box.
4
Find the Base vectors subsection. In the table, enter the following settings:
 
x
y
z
x1
cos(pi/4)
0
-sin(pi/4)
x3
sin(pi/4)
0
cos(pi/4)
Create a mapping operator that will be used to map the results between the submodels.
Identity Mapping 1 (idmap1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Identity Mapping.
2
In the Settings window for Identity Mapping, locate the Source Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 6 only.
Pressure Acoustics, Transient (actd)
Modify the Typical Wave Speed for Perfectly Matched Layers and Transient Solver Settings according to the fluid material and the frequency of the input signal. These settings are used to adjust the PML and the time-dependent solver settings.
1
In the Model Builder window, under Component 1 (comp1) click Pressure Acoustics, Transient (actd).
2
In the Settings window for Pressure Acoustics, Transient, locate the Typical Wave Speed for Perfectly Matched Layers section.
3
In the cref text field, type c0.
4
Locate the Transient Solver Settings section. In the Maximum frequency to resolve field enter f0.
Symmetry 1
1
In the Physics toolbar, click  Boundaries and choose Symmetry.
2
Select Boundaries 5 and 12 only.
Solid Mechanics (solid)
In the Model Builder window, under Component 1 (comp1) click Solid Mechanics (solid).
Piezoelectric Material 1
1
In the Physics toolbar, click  Domains and choose Piezoelectric Material.
2
Select Domain 3 only.
3
In the Settings window for Piezoelectric Material, locate the Coordinate System Selection section.
4
From the Coordinate system list, choose sys_PZT (sys2).
Mechanical Damping 1
1
In the Physics toolbar, click  Attributes and choose Mechanical Damping.
2
In the Settings window for Mechanical Damping, locate the Damping Settings section.
3
From the Damping type list, choose Rayleigh damping.
4
In the αdM text field, type alpha_dmp.
5
In the βdK text field, type beta_dmp.
Fixed Constraint 1
1
In the Physics toolbar, click  Boundaries and choose Fixed Constraint.
2
Select Boundaries 2 and 15 only.
Symmetry 1
1
In the Physics toolbar, click  Boundaries and choose Symmetry.
2
Select Boundaries 1 and 8 only.
Roller 1
1
In the Physics toolbar, click  Boundaries and choose Roller.
2
Select Boundary 9 only.
Electrostatics (es)
In the Model Builder window, under Component 1 (comp1) click Electrostatics (es).
Charge Conservation, Piezoelectric 1
1
In the Physics toolbar, click  Domains and choose Charge Conservation, Piezoelectric.
2
Select Domain 3 only.
Ground 1
1
In the Physics toolbar, click  Boundaries and choose Ground.
2
Select Boundary 9 only.
Electric Potential 1
1
In the Physics toolbar, click  Boundaries and choose Electric Potential.
2
Select Boundary 10 only.
3
In the Settings window for Electric Potential, locate the Electric Potential section.
4
In the V0 text field, type voltage_pulse_fct(t).
Add Multiphysics
1
In the Physics toolbar, click  Add Multiphysics to open the Add Multiphysics window.
2
Go to the Add Multiphysics window.
3
In the tree, select Acoustics>Acoustic-Structure Interaction>Acoustic-Piezoelectric Interaction, Transient.
4
Click Add to Component in the window toolbar.
5
In the Physics toolbar, click  Add Multiphysics to close the Add Multiphysics window.
Mesh 1
Free Triangular 1
1
In the Mesh toolbar, click  Boundary and choose Free Triangular.
2
Select Boundaries 3 and 10 only.
Size 1
1
Right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section. Select the Maximum element size check box.
5
In the associated text field, type lam0/6.
6
Click  Build Selected.
Swept 1
1
In the Mesh toolbar, click  Swept.
2
In the Settings window for Swept, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 3 only.
Distribution 1
1
Right-click Swept 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 2.
Swept 2
1
In the Mesh toolbar, click  Swept.
2
In the Settings window for Swept, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 1 only.
Distribution 1
1
Right-click Swept 2 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 4.
Swept 3
1
In the Mesh toolbar, click  Swept.
2
In the Settings window for Swept, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 2 only.
Size 1
1
Right-click Swept 3 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section. Select the Maximum element size check box.
5
In the associated text field, type lam0/6.
Swept 4
In the Mesh toolbar, click  Swept.
Distribution 1
1
Right-click Swept 4 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 6.
4
Click  Build All.
Component 1 (comp1)
Click the  Zoom Extents button in the Graphics toolbar.
Add Component
In the Model Builder window, right-click the root node and choose Add Component>3D.
Geometry 2
Create the geometry that contains the main pipe, the signal tube, and the receiver. As the transmitter, the receiver contains a piezoelectric transducer and a matched layer.
Cylinder 1 (cyl1)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type D/2.
4
In the Height text field, type L.
5
Locate the Axis section. From the Axis type list, choose x-axis.
6
Locate the Layers section. In the table, enter the following settings:
Layer name
Thickness (m)
Layer 1
0.5*D
7
Clear the Layers on side check box.
8
Select the Layers on bottom check box.
9
Select the Layers on top check box.
Cylinder 2 (cyl2)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type D_transducer/2.
4
In the Height text field, type L_transducer.
5
Locate the Position section. In the x text field, type L/2.
6
In the z text field, type -L_transducer/2.
Cylinder 3 (cyl3)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type D_transducer/2.
4
In the Height text field, type L_matching.
5
Locate the Position section. In the x text field, type L/2.
6
In the z text field, type L_transducer/2.
Cylinder 4 (cyl4)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type D_transducer/4.
4
In the Height text field, type L_piezo.
5
Locate the Position section. In the x text field, type L/2.
6
In the z text field, type L_transducer/2+L_matching.
Rotate 1 (rot1)
1
In the Geometry toolbar, click  Transforms and choose Rotate.
2
Select the objects cyl2, cyl3, and cyl4 only.
3
In the Settings window for Rotate, locate the Rotation section.
4
In the Angle text field, type alpha.
5
From the Axis type list, choose y-axis.
6
Locate the Point on Axis of Rotation section. In the x text field, type L/2.
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
From the Plane list, choose xz-plane.
Partition Objects 1 (par1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Partition Objects.
2
Click in the Graphics window and then press Ctrl+A to select all objects.
3
In the Settings window for Partition Objects, locate the Partition Objects section.
4
From the Partition with list, choose Work plane.
Delete Entities 1 (del1)
1
In the Model Builder window, right-click Geometry 2 and choose Delete Entities.
2
In the Settings window for Delete Entities, locate the Entities or Objects to Delete section.
3
From the Geometric entity level list, choose Domain.
4
On the object par1(1), select Domains 1, 3, and 5 only.
5
On the object par1(2), select Domain 1 only.
6
On the object par1(3), select Domain 1 only.
7
On the object par1(4), select Domain 1 only.
Form Composite Domains 1 (cmd1)
1
In the Geometry toolbar, click  Virtual Operations and choose Form Composite Domains.
2
On the object fin, select Domains 2–5 only.
3
In the Geometry toolbar, click  Build All.
4
Click the  Zoom Extents button in the Graphics toolbar.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in>Water, liquid.
4
Click Add to Component in the window toolbar.
5
In the tree, select Built-in>Lead Zirconate Titanate (PZT-5H).
6
Click Add to Component in the window toolbar.
7
In the tree, select Built-in>Acrylic plastic.
8
Click Add to Component in the window toolbar.
9
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
Lead Zirconate Titanate (PZT-5H) (mat5)
1
In the Model Builder window, under Component 2 (comp2)>Materials click Lead Zirconate Titanate (PZT-5H) (mat5).
2
Select Domain 4 only.
Acrylic plastic (mat6)
1
In the Model Builder window, click Acrylic plastic (mat6).
2
Select Domain 3 only.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select Acoustics>Ultrasound>Convected Wave Equation, Time Explicit (cwe).
4
Find the Physics interfaces in study subsection. In the table, clear the Solve check box for Study 1.
5
Click Add to Component 2 in the window toolbar.
6
In the tree, select Structural Mechanics>Solid Mechanics (solid).
7
In the table, clear the Solve check box for Study 1.
8
Click Add to Component 2 in the window toolbar.
9
In the tree, select AC/DC>Electric Fields and Currents>Electrostatics (es).
10
In the table, clear the Solve check box for Study 1.
11
Click Add to Component 2 in the window toolbar.
12
In the Home toolbar, click  Add Physics to close the Add Physics window.
Convected Wave Equation, Time Explicit (cwe)
1
In the Model Builder window, under Component 2 (comp2) click Convected Wave Equation, Time Explicit (cwe).
2
Select Domains 1, 2, and 5 only.
Solid Mechanics 2 (solid2)
1
In the Model Builder window, under Component 2 (comp2) click Solid Mechanics 2 (solid2).
2
Select Domains 3 and 4 only.
Electrostatics 2 (es2)
1
In the Model Builder window, under Component 2 (comp2) click Electrostatics 2 (es2).
2
Select Domain 4 only.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies>Time Dependent.
4
Find the Physics interfaces in study subsection. In the table, clear the Solve check boxes for Pressure Acoustics, Transient (actd), Solid Mechanics (solid), Electrostatics (es), Solid Mechanics 2 (solid2), and Electrostatics 2 (es2).
5
Find the Multiphysics couplings in study subsection. In the table, clear the Solve check boxes for Acoustic-Structure Boundary 1 (asb1) and Piezoelectric Effect 1 (pze1).
6
Click Add Study in the window toolbar.
7
Find the Physics interfaces in study subsection. In the table, clear the Solve check boxes for Pressure Acoustics, Transient (actd), Solid Mechanics (solid), Electrostatics (es), and Convected Wave Equation, Time Explicit (cwe).
8
Find the Multiphysics couplings in study subsection. In the table, clear the Solve check boxes for Acoustic-Structure Boundary 1 (asb1) and Piezoelectric Effect 1 (pze1).
9
Click Add Study in the window toolbar.
10
In the Home toolbar, click  Add Study to close the Add Study window.
Definitions (comp2)
Symmetry
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, locate the Input Entities section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundaries 2, 6, and 25 only.
5
In the Label text field, type Symmetry.
Source
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, locate the Input Entities section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 10 only.
5
In the Label text field, type Source.
Receiver
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, locate the Input Entities section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 16 only.
5
In the Label text field, type Receiver.
Walls
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, locate the Input Entities section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundaries 3, 4, 7–9, 11–13, 26, and 27 only.
5
In the Label text field, type Walls.
Absorbing Layer 1 (ab1)
1
In the Definitions toolbar, click  Absorbing Layer.
Add absorbing layers (sponge layers) to truncate the computational domain.
2
Select Domains 1 and 5 only.
Identity Mapping 2 (idmap2)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Identity Mapping.
2
In the Settings window for Identity Mapping, locate the Source Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 16 only.
Again, create a coordinate system specifying the piezoelectric material orientation.
sys_PZT
1
In the Definitions toolbar, click  Coordinate Systems and choose Base Vector System.
2
In the Settings window for Base Vector System, type sys_PZT in the Label text field.
3
Locate the Settings section. Find the Simplifications subsection. Select the Assume orthonormal check box.
4
Find the Base vectors subsection. In the table, enter the following settings:
 
x
y
z
x1
cos(pi/4)
0
-sin(pi/4)
x3
sin(pi/4)
0
cos(pi/4)
Convected Wave Equation, Time Explicit (cwe)
In the Model Builder window, under Component 2 (comp2) click Convected Wave Equation, Time Explicit (cwe).
Symmetry 1
1
In the Physics toolbar, click  Boundaries and choose Symmetry.
2
In the Settings window for Symmetry, locate the Boundary Selection section.
3
From the Selection list, choose Symmetry.
Acoustic Impedance 1
1
In the Physics toolbar, click  Boundaries and choose Acoustic Impedance.
2
Select Boundaries 1 and 28 only.
Acoustic Impedance 2
1
In the Physics toolbar, click  Boundaries and choose Acoustic Impedance.
On the receiver-fluid interface, specify the characteristic acoustic impedance of the matching layer.
2
In the Settings window for Acoustic Impedance, locate the Boundary Selection section.
3
From the Selection list, choose Receiver.
4
Locate the Acoustic Impedance section. In the Z text field, type Z_matching.
Normal Velocity 1
1
In the Physics toolbar, click  Boundaries and choose Normal Velocity.
On the transmitter-fluid interface, specify the velocity obtained from the transmitter submodel.
2
In the Settings window for Normal Velocity, locate the Boundary Selection section.
3
From the Selection list, choose Source.
4
Locate the Normal Velocity section. From the Type list, choose Velocity.
5
Specify the vb(t) vector as
comp1.idmap1(withsol('sol1', ut, setval(t, t)))
x
comp1.idmap1(withsol('sol1', vt, setval(t, t)))
y
comp1.idmap1(withsol('sol1', wt, setval(t, t)))
z
Solid Mechanics 2 (solid2)
In the Model Builder window, under Component 2 (comp2) click Solid Mechanics 2 (solid2).
Fixed Constraint 1
1
In the Physics toolbar, click  Boundaries and choose Fixed Constraint.
2
Select Boundaries 14 and 22 only.
Symmetry 1
1
In the Physics toolbar, click  Boundaries and choose Symmetry.
2
Select Boundaries 15 and 19 only.
Roller 1
1
In the Physics toolbar, click  Boundaries and choose Roller.
2
Select Boundary 21 only.
Boundary Load 1
1
In the Physics toolbar, click  Boundaries and choose Boundary Load.
2
In the Settings window for Boundary Load, locate the Boundary Selection section.
3
From the Selection list, choose Receiver.
4
Locate the Force section. From the Load type list, choose Pressure.
5
In the p text field, type idmap2(withsol('sol2', p2, setval(t, t))).
Piezoelectric Material 1
1
In the Physics toolbar, click  Domains and choose Piezoelectric Material.
2
Select Domain 4 only.
3
In the Settings window for Piezoelectric Material, locate the Coordinate System Selection section.
4
From the Coordinate system list, choose sys_PZT (sys4).
Mechanical Damping 1
1
In the Physics toolbar, click  Attributes and choose Mechanical Damping.
2
In the Settings window for Mechanical Damping, locate the Damping Settings section.
3
From the Damping type list, choose Rayleigh damping.
4
In the αdM text field, type alpha_dmp.
5
In the βdK text field, type beta_dmp.
Electrostatics 2 (es2)
In the Model Builder window, under Component 2 (comp2) click Electrostatics 2 (es2).
Ground 1
1
In the Physics toolbar, click  Boundaries and choose Ground.
2
Select Boundary 21 only.
Floating Potential 1
1
In the Physics toolbar, click  Boundaries and choose Floating Potential.
2
Select Boundary 20 only.
3
In the Settings window for Floating Potential, locate the Floating Potential section.
4
Select the Floating potential group check box.
Charge Conservation, Piezoelectric 1
1
In the Physics toolbar, click  Domains and choose Charge Conservation, Piezoelectric.
2
Select Domain 4 only.
Multiphysics
Piezoelectric Effect 2 (pze2)
In the Physics toolbar, click  Multiphysics Couplings and choose Domain>Piezoelectric Effect.
Mesh 2
Free Tetrahedral 1
1
In the Mesh toolbar, click  Free Tetrahedral.
2
In the Settings window for Free Tetrahedral, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 1, 2, and 5 only.
When modeling using physics interfaces that are based on the DG method, it is important to avoid small mesh elements, since they control the time steps taken by the solver. In order to avoid this use the Element Quality Optimization functionality available for the tetrahedral mesh. This step is very important.
5
Click to expand the Element Quality Optimization section. Select the Avoid too small elements check box.
6
From the Optimization level list, choose High.
Size
1
In the Model Builder window, click Size.
2
In the Settings window for Size, click to expand the Element Size Parameters section.
3
Locate the Element Size section. Click the Custom button.
4
Locate the Element Size Parameters section. In the Maximum element size text field, type lam0/1.5.
5
In the Minimum element size text field, type lam0/3.
6
In the Model Builder window, right-click Mesh 2 and choose Build All.
Mesh 3
In the Mesh toolbar, click  Add Mesh.
Free Triangular 1
1
In the Mesh toolbar, click  Boundary and choose Free Triangular.
2
Select Boundaries 17 and 20 only.
Size 1
1
Right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section. Select the Maximum element size check box.
5
In the associated text field, type lam0/6.
Swept 1
1
In the Mesh toolbar, click  Swept.
2
In the Settings window for Swept, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 4 only.
Distribution 1
1
Right-click Swept 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 2.
Swept 2
1
In the Mesh toolbar, click  Swept.
2
In the Settings window for Swept, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 3 only.
Distribution 1
1
Right-click Swept 2 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 4.
4
In the Model Builder window, right-click Mesh 3 and choose Build All.
Component 2 (comp2)
Click the  Zoom Extents button in the Graphics toolbar.
Study 1
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, locate the Study Settings section.
3
Clear the Generate default plots check box.
Transmitter
1
In the Model Builder window, under Study 1 click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, type Transmitter in the Label text field.
3
Locate the Study Settings section. In the Output times text field, type range(0,T0/5,30*T0).
4
Locate the Physics and Variables Selection section. In the table, clear the Solve for check box for Piezoelectric Effect 2 (pze2).
5
Click to expand the Mesh Selection section.
Study 2
1
In the Model Builder window, click Study 2.
2
In the Settings window for Study, locate the Study Settings section.
3
Clear the Generate default plots check box.
Propagation of Signal
1
In the Model Builder window, under Study 2 click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, type Propagation of Signal in the Label text field.
3
Locate the Study Settings section. In the Output times text field, type range(0,T0/5,30*T0).
4
Locate the Physics and Variables Selection section. In the table, clear the Solve for check box for Piezoelectric Effect 2 (pze2).
5
Click to expand the Values of Dependent Variables section. Find the Store fields in output subsection. From the Settings list, choose For selections.
6
Under Selections, click  Add.
Store the results on the structure-fluid interfaces and the symmetry plane only.
7
In the Add dialog box, in the Selections list, choose Symmetry, Source, and Receiver.
8
Click OK.
Study 3
1
In the Model Builder window, click Study 3.
2
In the Settings window for Study, locate the Study Settings section.
3
Clear the Generate default plots check box.
Receiver
1
In the Model Builder window, under Study 3 click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, type Receiver in the Label text field.
3
Locate the Study Settings section. In the Output times text field, type range(0,T0/5,30*T0).
4
From the Tolerance list, choose User controlled.
5
In the Relative tolerance text field, type 1e-4.
6
Locate the Mesh Selection section. In the table, enter the following settings:
Geometry
Mesh
Geometry 1
No mesh
Geometry 2
Mesh 3
Since this multiphysics problem involves a transient acoustics interface, the Transient Solver Settings defined for the acoustics interface will be automatically used when solving the coupled problem. This ensures optimal solver configuration for this piezoacoustic problem solved in the time domain.
Study 1
In the Home toolbar, click  Compute.
Study 2
1
Click  Compute.
The default solver settings generated for Study 3 will be different from those in Study 1. This is because the Pressure Acoustics, Transient interface is absent here. It is required to change the default solver settings to Manual and specify the time step explicitly.
Study 3
Solution 3 (sol3)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 3 (sol3) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, click to expand the Time Stepping section.
4
From the Steps taken by solver list, choose Manual.
5
In the Time step text field, type 1/(60*f0).
This expression is the same as the one generated in Study 1.
6
In the Study toolbar, click  Compute.
Results
Mirror 3D 1
1
In the Model Builder window, expand the Results node.
2
Right-click Results>Datasets and choose More 3D Datasets>Mirror 3D.
3
In the Settings window for Mirror 3D, locate the Plane Data section.
4
From the Plane list, choose zx-planes.
Create a plot that shows the propagating acoustic signal induced by the driving voltage applied to the transmitter. The result should look as in Figure 2.
Transmitter
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type Transmitter in the Label text field.
3
Locate the Data section. From the Time (s) list, choose 2.4E-6.
Surface 1
1
Right-click Transmitter and choose Surface.
2
In the Settings window for Surface, locate the Coloring and Style section.
3
From the Color table list, choose Wave.
4
Select the Symmetrize color range check box.
Surface 2
1
In the Model Builder window, right-click Transmitter and choose Surface.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Mirror 3D 1.
4
Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Solid Mechanics>Displacement>solid.disp - Displacement magnitude - m.
5
Locate the Data section. From the Time (s) list, choose 2.4E-6.
Deformation 1
1
Right-click Surface 2 and choose Deformation.
2
In the Settings window for Deformation, locate the Scale section.
3
Select the Scale factor check box.
4
In the associated text field, type 1e3.
Transmitter
1
Click the  Zoom Extents button in the Graphics toolbar.
2
In the Model Builder window, click Transmitter.
3
In the Transmitter toolbar, click  Plot.
Next, create a plot showing the propagation of the acoustic signal through the main pipe. For selected times, the acoustic signal profile can be seen in Figure 3.
Signal Propagation
1
In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Dataset list, choose Study 2/Solution 2 (4) (sol2).
4
From the Time (s) list, choose 2.4E-6.
5
In the Label text field, type Signal Propagation.
Surface 1
1
Right-click Signal Propagation and choose Surface.
2
In the Settings window for Surface, locate the Coloring and Style section.
3
From the Color table list, choose Wave.
4
Select the Symmetrize color range check box.
5
Click to expand the Quality section. Modify the Resolution and increase the Element refinement to represent the solution properly. This is because the shape functions used for the convected wave equation interface are 4th order per default.
6
From the Resolution list, choose Custom.
7
In the Element refinement text field, type 6.
Selection 1
1
Right-click Surface 1 and choose Selection.
2
In the Settings window for Selection, locate the Selection section.
3
From the Selection list, choose Symmetry.
4
In the Signal Propagation toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Plot the mechanical deformation of the receiver caused by the incoming acoustic signal. The deformation is depicted in Figure 4.
Receiver
1
In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, type Receiver in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 3/Solution 3 (6) (sol3).
4
From the Time (s) list, choose 1E-5.
Surface 1
1
Right-click Receiver and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 2 (comp2)>Solid Mechanics 2>Displacement>solid2.disp - Displacement magnitude - m.
Deformation 1
1
Right-click Surface 1 and choose Deformation.
2
In the Settings window for Deformation, locate the Scale section.
3
Select the Scale factor check box.
4
In the associated text field, type 1e3.
5
Click the  Zoom Extents button in the Graphics toolbar.
6
In the Receiver toolbar, click  Plot.
The last plot will show the input voltage applied to the transmitter and the output voltage signal read on the receiver (see Figure 5).
Sent and Received Signals
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Sent and Received Signals in the Label text field.
3
Locate the Data section. From the Dataset list, choose None.
Point Graph 1
1
Right-click Sent and Received Signals and choose Point Graph.
2
In the Settings window for Point Graph, locate the Data section.
3
From the Dataset list, choose Study 1/Solution 1 (1) (sol1).
4
Select Point 4 only.
5
Click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Electrostatics>Electric>V - Electric potential - V.
6
Click to expand the Legends section. Select the Show legends check box.
7
From the Legends list, choose Manual.
8
In the table, enter the following settings:
Legends
Driving voltage
9
Locate the Data section. From the Time selection list, choose Interpolated.
10
Click  Range.
11
In the Range dialog box, choose Number of values from the Entry method list.
12
In the Start text field, type 0.
13
In the Stop text field, type 30*T0.
14
In the Number of values text field, type 2001.
15
Click Replace.
Point Graph 2
1
In the Model Builder window, right-click Sent and Received Signals and choose Point Graph.
2
In the Settings window for Point Graph, locate the Data section.
3
From the Dataset list, choose Study 3/Solution 3 (6) (sol3).
4
Select Point 17 only.
5
Click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 2 (comp2)>Electrostatics 2>Floating potentials>es2.fp1.V - Floating potential - V.
6
Locate the Legends section. Select the Show legends check box.
7
From the Legends list, choose Manual.
8
In the table, enter the following settings:
Legends
Electric potential on receiver
9
Locate the Data section. From the Time selection list, choose Interpolated.
10
Click  Range.
11
In the Range dialog box, choose Number of values from the Entry method list.
12
In the Start text field, type 0.
13
In the Stop text field, type 30*T0.
14
In the Number of values text field, type 2001.
15
Click Replace.
Sent and Received Signals
1
In the Model Builder window, click Sent and Received Signals.
2
In the Settings window for 1D Plot Group, locate the Plot Settings section.
3
Select the Two y-axes check box.
4
In the table, select the Plot on secondary y-axis check box for Point Graph 2.
5
Locate the Legend section. From the Position list, choose Upper middle.
6
Locate the Axis section. Select the Manual axis limits check box.
7
In the y minimum text field, type -51.
8
In the y maximum text field, type 51.
9
In the Secondary y minimum text field, type -1.2.
10
In the Secondary y maximum text field, type 1.2.
11
In the Sent and Received Signals toolbar, click  Plot.