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Duct with Right-Angled Bend
Introduction
In this tutorial, the acoustic behavior of a duct or waveguide with a right-angled bend is analyzed. The model uses port boundary conditions at the inlet and outlet. The ports can capture and treat nonplane propagating modes in waveguides, extending the analysis above the first cutoff frequency. The transmission loss and the scattering coefficients of the system are determined.
Model Definition
The geometry of the waveguide system analyzed in this model is depicted in Figure 1. It consists of a duct with a right-angled bend in the middle. On the outlet branch, there is also a small dent. The dent is present in order to excite modes that have components in the duct height (the z direction). The width of the duct is W and the height is H; these are parameters that can be changed. A 2D variant of this tutorial model that studies scattering in a 2D bend is described in Ref. 1.
Figure 1: Waveguide geometry consisting of two sections connected by a right-angled bend. A small dent is present on the outlet section.
The model is solved for frequencies in the range from 50 to 1300 Hz. In that range five modes exist: the plane-wave mode (0,0), the first long-edge mode (1,0), the first short-edge (0,1), the first mixed mode (1,1), and the second long-edge mode (2,0). Their cutoff frequencies are given in Table 1 below (these are derived in the results analysis steps of the model using built-in variables). The mode shapes are depicted in the results section in Figure 6.
Table 1: Mode cutoff frequencies.
Mode (m,n)
Cutoff frequency
(0,0)
0.0 Hz
(1,0)
572.0 Hz
(0,1)
858.0 Hz
(1,1)
1031.2 Hz
(2,0)
1144.0 Hz
In order to capture these modes and ensure good nonreflecting behavior at the waveguide inlet and outlet, four Port boundary conditions are added at each end, each port capturing a specific mode. When solving the model, parts of the solution are the scattering coefficients Si1, where the subscript 1 refers to the fact that the system is excited at port 1 (plane wave incidence at the inlet). This means that on the inlet or outlet boundaries, the combination of ports defines the total acoustic field (sum of incident and outgoing waves, propagating and nonpropagating) as
where the summation “i” is over all ports on the given boundary “bnd” (for example 1, 2, 3, 4, and 5 on the inlet), Ain is the amplitude of the incident field (1 Pa in this model), and pi is the mode shape of the ith port. The mode shape pi is normalized to have a unit maximum amplitude.
A benefit of using the Port boundary condition is that important postprocessing variables are readily defined on each port. This includes fields like the total pressure, intensity, and normal velocity, and also global (integrated) variables like the incoming or outgoing power or average pressure. These quantities can in general be difficult to compute when complex nonplane propagation occurs. In this model, the transmission loss TL of the system is calculated as
where the sum is over all the ports at the outlet. The incident power is simply given by the variable acpr.port1.P_in and the sum of the outgoing power as acpr.port6.P_out+acpr.port7.P_out+acpr.port8.P_out+acpr.port9.P_out+acpr.port10.P_out.
Results and Discussion
The absolute values of the scattering coefficients are depicted in Figure 2 below. The normalized power of the outgoing modes, through the ports, is presented in Figure 3. The graphs shows the cutoff /on of the different modes which is in accordance with the values given in Table 1. Note that the cutoff/on as depicted when plotting does not occur as sharply as when visualizing the power. The reason is that pure evanescent modes will contribute to the total field at the port (giving a nonzero S-parameter); while these modes do not carry any real power as they are not propagating.
The transmission loss of the system is depicted in Figure 4. Isosurface plots of the pressure inside the duct is depicted in Figure 5 at four different frequencies, each representing values when one more mode is propagating in the system. The first image at the upper left represents plane wave propagation and so forth. Finally, the five normalized mode shapes are depicted in Figure 6.
Figure 2: Absolute value of the scattering coefficients as function of frequency.
Figure 3: Normalized power of the outgoing modes as function of frequency.
Figure 4: Transmission loss TL through the waveguide system.
Figure 5: Pressure isosurfaces evaluated at four different frequencies.
Figure 6: Normalized mode shapes for the five ports.
Reference
1. T. Graf and J. Pan, “Determination of the complex acoustic scattering matrix of a right-angled duct,” J. Acoust. Soc. Am., vol. 134, pp. 292–299, 2013.
Application Library path: Acoustics_Module/Tutorials,_Pressure_Acoustics/duct_right_angled_bend
Modeling Instructions
This section contains the modeling instructions for the Duct with Right-Angled Bend model. They are followed by the Geometry Sequence Instructions section.
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Acoustics > Pressure Acoustics > Pressure Acoustics, Frequency Domain (acpr).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Frequency Domain.
6
Click  Done.
Geometry 1
To save some time, import the geometry sequence from a file. The instructions for setting up the geometry can be found in the Geometry Sequence Instructions section at the bottom of this document.
1
In the Geometry toolbar, click Insert Sequence and choose Insert Sequence.
2
Browse to the model’s Application Libraries folder and double-click the file duct_right_angled_bend_geom_sequence.mph.
3
In the Geometry toolbar, click  Build All.
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
In the table, enter the following settings:
Name
Expression
Value
Description
f_max
1300[Hz]
1300 Hz
Maximum study frequency
lambda_min
343[m/s]/f_max
0.26385 m
Minimum wavelength
Definitions
Create selections for the inlet and outlet of the duct.
Inlet
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Inlet in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundary 1 only.
Outlet
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Outlet in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Click the  Wireframe Rendering button in the Graphics toolbar. This simplifies seeing and selecting the outlet boundary.
5
Select Boundary 12 only.
6
Click the  Wireframe Rendering button in the Graphics toolbar.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in > Air.
4
Click the Add to Component button in the window toolbar.
5
In the Materials toolbar, click  Add Material to close the Add Material window.
Proceed to setting up the physics by adding the Port boundary conditions to the inlet and the outlet. Use the Node Group feature to group the conditions at the inlet and outlet, respectively. This gives a better overview of the model setup.
Add five port conditions at each end of the duct (the waveguide structure). These will capture all propagating modes up to the studied frequency of 1200 Hz. In the postprocessing part of this model you will set up and see a list of the cutoff frequencies of the captured modes. It is always possible to use the Pressure Acoustics, Boundary Mode interface to study and analyze propagating and nonpropagating modes in a waveguide cross section. See, for example, the eigenmodes_in_muffler model in the Automotive application library.
Pressure Acoustics, Frequency Domain (acpr)
Inlet Ports
1
In the Model Builder window, under Component 1 (comp1) right-click Pressure Acoustics, Frequency Domain (acpr) and choose Node Group.
2
In the Settings window for Group, type Inlet Ports in the Label text field.
Port 1
1
In the Physics toolbar, click  Boundaries and choose Port.
2
In the Settings window for Port, locate the Boundary Selection section.
3
From the Selection list, choose Inlet.
4
Locate the Port Properties section. From the Type of port list, choose Rectangular.
Use the rectangular (analytical) option that applies to this geometry of a rectangular waveguide with sound hard boundaries. Note that the first port condition that you add has the incident mode option turned on per default, whereas the following are off per default.
5
Locate the Incident Mode Settings section. In the Apin text field, type 1.
Port 2
1
In the Physics toolbar, click  Boundaries and choose Port.
2
In the Settings window for Port, locate the Boundary Selection section.
3
From the Selection list, choose Inlet.
4
Locate the Port Properties section. From the Type of port list, choose Rectangular.
5
Locate the Port Mode Settings section. In the m text field, type 1.
Port 3
1
In the Physics toolbar, click  Boundaries and choose Port.
2
In the Settings window for Port, locate the Boundary Selection section.
3
From the Selection list, choose Inlet.
4
Locate the Port Properties section. From the Type of port list, choose Rectangular.
5
Locate the Port Mode Settings section. In the n text field, type 1.
Port 4
1
In the Physics toolbar, click  Boundaries and choose Port.
2
In the Settings window for Port, locate the Boundary Selection section.
3
From the Selection list, choose Inlet.
4
Locate the Port Properties section. From the Type of port list, choose Rectangular.
5
Locate the Port Mode Settings section. In the m text field, type 1.
6
In the n text field, type 1.
Port 5
1
In the Physics toolbar, click  Boundaries and choose Port.
2
In the Settings window for Port, locate the Boundary Selection section.
3
From the Selection list, choose Inlet.
4
Locate the Port Properties section. From the Type of port list, choose Rectangular.
5
Locate the Port Mode Settings section. In the m text field, type 2.
6
In the n text field, type 0.
Outlet Ports
1
In the Model Builder window, right-click Pressure Acoustics, Frequency Domain (acpr) and choose Node Group.
2
In the Settings window for Group, type Outlet Ports in the Label text field.
Port 6
1
In the Physics toolbar, click  Boundaries and choose Port.
2
In the Settings window for Port, locate the Boundary Selection section.
3
From the Selection list, choose Outlet.
4
Locate the Port Properties section. From the Type of port list, choose Rectangular.
Port 7
1
In the Physics toolbar, click  Boundaries and choose Port.
2
In the Settings window for Port, locate the Boundary Selection section.
3
From the Selection list, choose Outlet.
4
Locate the Port Properties section. From the Type of port list, choose Rectangular.
5
Locate the Port Mode Settings section. In the m text field, type 1.
Port 8
1
In the Physics toolbar, click  Boundaries and choose Port.
2
In the Settings window for Port, locate the Boundary Selection section.
3
From the Selection list, choose Outlet.
4
Locate the Port Properties section. From the Type of port list, choose Rectangular.
5
Locate the Port Mode Settings section. In the n text field, type 1.
Port 9
1
In the Physics toolbar, click  Boundaries and choose Port.
2
In the Settings window for Port, locate the Boundary Selection section.
3
From the Selection list, choose Outlet.
4
Locate the Port Properties section. From the Type of port list, choose Rectangular.
5
Locate the Port Mode Settings section. In the m text field, type 1.
6
In the n text field, type 1.
Port 10
1
In the Physics toolbar, click  Boundaries and choose Port.
2
In the Settings window for Port, locate the Boundary Selection section.
3
From the Selection list, choose Outlet.
4
Locate the Port Properties section. From the Type of port list, choose Rectangular.
5
Locate the Port Mode Settings section. In the m text field, type 2.
6
In the n text field, type 0.
Mesh 1
In this model, the mesh is set up manually. Proceed by directly adding the desired mesh component.
Use a swept mesh to reduce the solving time.
Swept 1
In the Mesh toolbar, click  Swept.
Size
1
In the Model Builder window, click 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. In the Maximum element size text field, type lambda_min/5.
In general, 5 to 6 second-order elements per wavelength are needed to resolve the waves. For more details, see Meshing (Resolving the Waves) in the Acoustics Module User’s Guide. In this model, we use 5 elements per wavelength.
Swept 1
1
In the Model Builder window, click Swept 1.
2
In the Settings window for Swept, click to expand the Source Faces section.
3
Select Boundaries 4 and 9–11 only.
4
Click to expand the Destination Faces section. Select Boundary 3 only.
5
Locate the Mesh Generation section. From the Elements list, choose Prisms.
6
Click  Build All.
Study 1
Step 1: Frequency Domain
1
In the Model Builder window, under Study 1 click Step 1: Frequency Domain.
2
In the Settings window for Frequency Domain, locate the Study Settings section.
3
Click  Range.
4
In the Range dialog, choose ISO preferred frequencies from the Entry method list.
5
In the Start frequency text field, type 50.
6
In the Stop frequency text field, type f_max.
7
From the Interval list, choose 1/24 octave.
8
Click Replace.
9
In the Study toolbar, click  Compute.
Results
Take a look at the first three default plots that are generated automatically. They represent the acoustic pressure, the sound pressure level, and isocontours of the pressure, respectively. Change the Parameter value (freq (Hz)) in the plots to change the evaluation frequency, for example, to 400 Hz.
Acoustic Pressure (acpr)
1
In the Model Builder window, under Results click Acoustic Pressure (acpr).
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Parameter value (freq (Hz)) list, choose 400.
Sound Pressure Level (acpr)
1
In the Model Builder window, click Sound Pressure Level (acpr).
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Parameter value (freq (Hz)) list, choose 400.
Acoustic Pressure, Isosurfaces (acpr)
1
In the Model Builder window, click Acoustic Pressure, Isosurfaces (acpr).
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Parameter value (freq (Hz)) list, choose 400.
The isosurface plots of the pressure at four different frequencies are also depicted in Figure 5.
Scattering Coefficients
Now proceed and plot the scattering coefficient to reproduce Figure 2.
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Scattering Coefficients in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Label.
4
Locate the Legend section. From the Position list, choose Upper left.
Global 1
1
Right-click Scattering Coefficients and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
abs(acpr.S11)
1
abs(S11)
abs(acpr.S21)
1
abs(S21)
abs(acpr.S31)
1
abs(S31)
abs(acpr.S41)
1
abs(S41)
abs(acpr.S51)
1
abs(S51)
abs(acpr.S61)
1
abs(S61)
abs(acpr.S71)
1
abs(S71)
abs(acpr.S81)
1
abs(S81)
abs(acpr.S91)
1
abs(S91)
abs(acpr.S10_1)
1
abs(S10_1)
4
In the Scattering Coefficients toolbar, click  Plot.
Transmission Loss
Now plot the transmission loss to reproduce Figure 4.
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Transmission Loss in the Label text field.
3
Locate the Title section. From the Title type list, choose Label.
4
Locate the Plot Settings section.
5
Select the y-axis label checkbox. In the associated text field, type Transmission Loss (dB).
6
Locate the Legend section. From the Position list, choose Upper left.
Octave Band 1
1
In the Transmission Loss toolbar, click  More Plots and choose Octave Band.
2
In the Settings window for Octave Band, locate the Selection section.
3
From the Geometric entity level list, choose Global.
4
Locate the y-Axis Data section. In the Expression text field, type acpr.port1.P_in/(acpr.port6.P_out+acpr.port7.P_out+acpr.port8.P_out+acpr.port9.P_out+acpr.port10.P_out).
This expression gives the ratio of the incoming power (the source at Port 1) and the total outgoing power at the outlet (the sum of the outgoing power of all modes). Note that in general this is not a straightforward quantity to compute for the nonplane modes. Thus, using the built-in variables for the power (acpr.port4.P_out and so on) greatly simplifies postprocessing.
5
From the Expression type list, choose Transfer function.
6
Locate the Plot section. From the Quantity list, choose Continuous power spectral density.
7
Click to expand the Legends section. Select the Show legends checkbox.
8
From the Legends list, choose Manual.
9
In the table, enter the following settings:
Legends
Transmission Loss - Continuous
10
In the Transmission Loss toolbar, click  Plot.
Octave Band 2
1
Right-click Octave Band 1 and choose Duplicate.
2
In the Settings window for Octave Band, locate the Plot section.
3
From the Quantity list, choose Band average power spectral density.
4
From the Band type list, choose 1/3 octave.
5
Click to expand the Coloring and Style section. From the Type list, choose Outline.
6
Locate the Legends section. In the table, enter the following settings:
Legends
Transmission Loss - 1/3 Octave Bands
7
In the Transmission Loss toolbar, click  Plot.
Surface 1
Now, plot the normalized propagation modes to reproduce Figure 6.
1
In the Model Builder window, expand the Results > Datasets node.
2
Right-click Results > Datasets and choose Surface.
3
Select Boundary 1 only.
Normalized Mode Shapes
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Normalized Mode Shapes in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Label.
Surface 1
1
Right-click Normalized Mode Shapes 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 1 (comp1) > Pressure Acoustics, Frequency Domain > Ports > Port 1 > acpr.port1.pn - Normalized mode pressure - Pa.
3
In the Normalized Mode Shapes toolbar, click  Plot.
4
Locate the Coloring and Style section. From the Color table list, choose Wave.
5
From the Scale list, choose Linear symmetric.
Surface 2
1
Right-click Surface 1 and choose Duplicate.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type acpr.port2.pn.
4
Click to expand the Inherit Style section. From the Plot list, choose Surface 1.
Transformation 1
1
Right-click Surface 2 and choose Transformation.
2
In the Settings window for Transformation, locate the Transformation section.
3
In the x text field, type 1.1*W.
4
In the Normalized Mode Shapes toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Surface 3
1
In the Model Builder window, under Results > Normalized Mode Shapes right-click Surface 2 and choose Duplicate.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type acpr.port3.pn.
Transformation 1
1
In the Model Builder window, expand the Surface 3 node, then click Transformation 1.
2
In the Settings window for Transformation, locate the Transformation section.
3
In the x text field, type 0.
4
In the y text field, type -1.1*H.
5
In the Normalized Mode Shapes toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar.
Surface 4
1
In the Model Builder window, under Results > Normalized Mode Shapes right-click Surface 3 and choose Duplicate.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type acpr.port4.pn.
Transformation 1
1
In the Model Builder window, expand the Surface 4 node, then click Transformation 1.
2
In the Settings window for Transformation, locate the Transformation section.
3
In the x text field, type 1.1*W.
4
In the Normalized Mode Shapes toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Surface 5
1
In the Model Builder window, under Results > Normalized Mode Shapes right-click Surface 4 and choose Duplicate.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type acpr.port5.pn.
Transformation 1
1
In the Model Builder window, expand the Surface 5 node, then click Transformation 1.
2
In the Settings window for Transformation, locate the Transformation section.
3
In the x text field, type 2.2*W.
4
In the y text field, type 0.
5
In the Normalized Mode Shapes toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar.
Cutoff Frequencies
1
In the Results toolbar, click  Evaluation Group.
2
In the Settings window for Evaluation Group, type Cutoff Frequencies in the Label text field.
3
Locate the Data section. From the Parameter selection (freq) list, choose First.
Global Evaluation 1
1
Right-click Cutoff Frequencies and choose Global Evaluation.
2
In the Settings window for Global Evaluation, click Replace Expression in the upper-right corner of the Expressions section. From the menu, choose Component 1 (comp1) > Pressure Acoustics, Frequency Domain > Ports > Port 1 > acpr.port1.fc - Mode cutoff frequency - Hz.
Modify the first description to reflect which mode it refers to and then add the other cutoff frequency variables. These values are also discussed in the results section of the model description.
3
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
acpr.port1.fc
Hz
Mode (0,0) cutoff frequency
acpr.port2.fc
Hz
Mode (1,0) cutoff frequency
acpr.port3.fc
Hz
Mode (0,1) cutoff frequency
acpr.port4.fc
Hz
Mode (1,1) cutoff frequency
acpr.port5.fc
Hz
Mode (2,0) cutoff frequency
4
In the Cutoff Frequencies toolbar, click  Evaluate.
Normalized Port Outgoing Power
Now, plot the normalized power of the outgoing modes.
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Normalized Port Outgoing Power in the Label text field.
3
Locate the Title section. From the Title type list, choose Label.
4
Locate the Legend section. From the Position list, choose Middle left.
Global 1
1
Right-click Normalized Port Outgoing Power and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
acpr.port1.P_out/acpr.port1.P_in
1
Port 1
acpr.port2.P_out/acpr.port1.P_in
1
Port 2
acpr.port3.P_out/acpr.port1.P_in
1
Port 3
acpr.port4.P_out/acpr.port1.P_in
1
Port 4
acpr.port5.P_out/acpr.port1.P_in
1
Port 5
Global 2
1
Right-click Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
acpr.port6.P_out/acpr.port1.P_in
1
Port 6
acpr.port7.P_out/acpr.port1.P_in
1
Port 7
acpr.port8.P_out/acpr.port1.P_in
1
Port 8
acpr.port9.P_out/acpr.port1.P_in
1
Port 9
acpr.port10.P_out/acpr.port1.P_in
1
Port 10
4
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.
5
From the Color list, choose Cycle (reset).
6
In the Normalized Port Outgoing Power toolbar, click  Plot.
Geometry Sequence 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
In the table, enter the following settings:
Name
Expression
Value
Description
L
120[cm]
1.2 m
Waveguide length
W
30[cm]
0.3 m
Waveguide width
H
20[cm]
0.2 m
Waveguide height
Rcurv
8[cm]
0.08 m
Bend curvature radius
Rcyl
20[cm]
0.2 m
Cylinder radius (for dent)
Add Component
In the Home toolbar, click  Add Component and choose 3D.
Geometry 1
Work Plane 1 (wp1)
In the Geometry toolbar, click  Work Plane.
Work Plane 1 (wp1) > Rectangle 1 (r1)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type L.
4
In the Height text field, type W.
Work Plane 1 (wp1) > Rectangle 2 (r2)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type W.
4
In the Height text field, type L.
5
Locate the Position section. In the xw text field, type L-W.
Work Plane 1 (wp1) > Union 1 (uni1)
1
In the Work Plane toolbar, click  Booleans and Partitions and choose Union.
2
Click in the Graphics window and then press Ctrl+A to select both objects.
3
In the Settings window for Union, locate the Union section.
4
Clear the Keep interior boundaries checkbox.
5
Click  Build Selected.
Work Plane 1 (wp1) > Fillet 1 (fil1)
1
In the Work Plane toolbar, click  Fillet.
2
On the object uni1, select Point 4 only.
3
In the Settings window for Fillet, locate the Radius section.
4
In the Radius text field, type Rcurv.
Work Plane 1 (wp1) > Fillet 2 (fil2)
1
In the Work Plane toolbar, click  Fillet.
2
On the object fil1, select Point 7 only.
3
In the Settings window for Fillet, locate the Radius section.
4
In the Radius text field, type 1.2*Rcurv.
5
Click  Build Selected.
6
Click the  Zoom Extents button in the Graphics toolbar.
Extrude 1 (ext1)
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 right-click Work Plane 1 (wp1) and choose Extrude.
2
In the Settings window for Extrude, locate the Distances section.
3
In the table, enter the following settings:
Distances (m)
H
4
Click  Build Selected.
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 Rcyl.
4
Locate the Position section. In the x text field, type L/2.
5
In the y text field, type L/2.
6
In the z text field, type H+0.8*Rcyl.
7
Locate the Axis section. From the Axis type list, choose x-axis.
Difference 1 (dif1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Difference.
2
Select the object ext1 only.
3
In the Settings window for Difference, locate the Difference section.
4
Click to select the  Activate Selection toggle button for Objects to subtract.
5
Select the object cyl1 only.
6
Click  Build All Objects.
7
Click the  Zoom Extents button in the Graphics toolbar.