Sensitivity Analysis of a Truss Tower

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Sensitivity Analysis of a Truss Tower

Sensitivity analysis is an efficient way of computing the gradient of an objective function with respect to many control variables. In this example, the pitch and yaw in the top of a truss tower are used as objective functions. The sensitivities of these angles to changes in the individual bar diameters are then computed.

Model Definition

The truss geometry consists of a unit cell that is repeated fives times, see Figure 1. The tower is made from structural steel.

Figure 1: The geometry of the tower.

The four points in the bottom are pinned, while the top four points are subjected to two load cases: bending and torsion.

The bending load case is created by subjecting each of the top four points to a force of 10 kN in the x direction.

The torsion load case is created by subjecting the points to a force of 10 kN in the xy-plane, with a direction that is orthogonal to the vector going from the center of the tower to the point.

Using straightforward trigonometry, the tilt and yaw angles are computed by considering the displacement of the top four corners.

Results and Discussion

Figure 2 shows the tilt sensitivity for the bending load case to the left and the torsion load case to the right, respectively. The tower does not tilt in response to torsion, and looking at the color scale one can see that this property is not sensitive to changes in the bar diameters. As one might expect, the lower vertical bars should be reinforced to reduce tilt in response to bending.

Figure 2: The tilt sensitivity of the tower is plotted for both the bending and the torsion load case.

Similarly, Figure 3 shows the yaw sensitivity. The tower does not yaw in response to bending, but it can be made to do so by strengthening and weakening some of the diagonal bars. As one would expect, the diagonal bars (at all heights) are the ones that need higher stiffness to reduce yaw in response to torsion.

Figure 3: The yaw sensitivity of the tower is plotted for both the bending and the torsion load case.

Notes About the COMSOL Implementation

The bar diameters are coupled to the sensitivity analysis by defining a control variable field on all edges. This is then used for the cross-sectional data in the truss interface.

The mast has 134 bars, but in this case the use of adjoint sensitivity doubles the computational cost compared to just solving for the displacement field. This model is linear, but for a nonlinear problem the sensitivity analysis only adds the cost of an extra nonlinear iteration.

Adjoint sensitivity analysis is the foundation of gradient-based optimization. It is supported for stationary (including frequency domain) and transient solvers. For more information, see Theory for the Sensitivity Interface in the Optimization Module User’s Guide.

Structural_Mechanics_Module/Sensitivity_and_Optimization/tower_sensitivity

Modeling Instructions

From the menu, choose .

In the window, click

.

1

In the window, click

.

2

In the tree, select .

3

4

In the tree, select .

5

6

7

In the tree, select .

8

Create the geometry. To simplify this step, insert a prepared geometry sequence.

1

In the toolbar, click

.

2

Browse to the model’s Application Libraries folder and double-click the file tower_sensitivity_geom_sequence.mph.

3

In the toolbar, click

.

4

button in the toolbar.

5

In the window, collapse the node.

1

In the toolbar, click

to open the window.

2

Go to the window.

3

In the tree, select .

4

Click in the window toolbar.

5

In the toolbar, click

to close the window.

Component 1 (comp1)

Add a cylindrical coordinate system for imposing the twisting load case.

Cylindrical System 2 (sys2)

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

Find the subsection. In the table, enter the following settings:


x (m)	y (m)	z (m)
Lx/2	Ly/2	0

Sensitivity (sens)

Control Variable Field 1

1

In the window, under right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. In the text field, type Abar.

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In the text field, type 1.

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Locate the section. From the list, choose .

7

Find the subsection. From the list, choose .

Global Definitions

Add the diameters of the bars to the list of parameters.

1

In the window, under click .

2

In the window for , locate the section.

3

In the table, enter the following settings:


Name	Expression	Value	Description
d1	1[cm]	0.01 m	Vertical bar diameter
d2	5[mm]	0.005 m	Diagonal and horizontal bar diameter

1

In the window, under click .

2

In the window for , locate the section.

3

From the list, choose .

Cross-Section Data 1

1

In the window, under click .

2

In the window for , locate the section.

3

In the A text field, type pi/4*d1^2*Abar.

Cross-Section Data 2

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. In the A text field, type pi/4*d2^2*Abar.

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. Specify the FP vector as


1[kN]	x
0	y
0	z

5

In the toolbar, click

and choose .

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. From the list, choose .

5

Locate the section. Specify the FP vector as


0	r
1[kN]	phi
0	a

6

In the toolbar, click

and choose .

Add an feature to compute the tilt and yaw angles of the truss tower.

Average Rotation 1

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

Global Definitions

Load Group: Bending

1

In the window, under click .

2

In the window for , type Load Group: Bending in the text field.

3

In the text field, type Bending.

Load Group: Torsion

1

In the window, under click .

2

In the window for , type Load Group: Torsion in the text field.

3

In the text field, type Torsion.

Tilt Sensitivity

1

In the window, click .

2

In the window for , type Tilt Sensitivity in the text field.

3

Locate the section. Clear the check box.

4

button in the toolbar.

5

In the dialog box, in the tree, select the check box for the node .

6

Click to enable the study step.

1

In the toolbar, click

.

2

In the window for , locate the section.

3

In the table, enter the following settings:


Expression	Description	Evaluate for
comp1.truss.avgr1.thY	Tilt angle	Stationary

Step 1: Stationary

1

In the window, click .

2

In the window for , click to expand the section.

3

Select the check box.

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twice.

5

In the table, enter the following settings:


Load case	Bending	Weight	Torsion	Weight
Bending	√	1.0		1.0
Torsion		1.0	√	1.0

6

In the toolbar, click

.

Global Evaluation 1

1

In the window, expand the node.

2

Right-click and choose .

3

In the window for , locate the section.

4

In the table, enter the following settings:


Expression	Unit	Description
truss.avgr1.thY	deg	Tilt angle
truss.avgr1.thZ	deg	Yaw angle

5

The tower only tilts in response to bending and yaws in response to torsion.

Tilt Sensitivity

1

In the toolbar, click

.

2

In the window for , type Tilt Sensitivity in the text field.

3

Click to expand the section. From the list, choose .

1

Right-click and choose .

2

In the window for , locate the section.

3

In the text field, type fsens(Abar).

4

Locate the section. From the list, choose .

5

Select the check box.

6

Click to expand the section. From the list, choose .

The tower does not tilt in response to torsion and the color scale tells that this property is not sensitive to the bar diameters.

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

From the list, choose .

5

Locate the section. From the list, choose .

1

Right-click and choose .

2

In the window for , locate the section.

3

In the text field, type -4.

4

In the text field, type 0.

5

In the text field, type 0.

6

Locate the section. Select the check box.

7

In the associated text field, type 1.

Tilt Sensitivity

In the window, click .

1

In the toolbar, click

and choose .

2

In the window for , click in the upper-right corner of the section. From the menu, choose .

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

From the list, choose .

1

Right-click and choose .

2

In the window for , locate the section.

3

In the text field, type -4.

4

In the text field, type 0.

5

In the text field, type 0.

6

Locate the section. Select the check box.

7

In the associated text field, type 1.

Tilt Sensitivity

1

In the window, click .

2

In the window for , locate the section.

3

From the list, choose .

4

button in the toolbar.

5

In the toolbar, click

.

As one might expect, the lower vertical bars should be made stiffer to reduce tilt in response to bending.

Add a new study to investigate the yaw sensitivity.

1

In the toolbar, click

to open the window.

2

Go to the window.

3

Find the subsection. In the tree, select .

4

Click in the window toolbar.

5

In the toolbar, click

to close the window.

Step 1: Stationary

1

In the window for , locate the section.

2

Select the check box.

3

twice.

4

In the table, enter the following settings:


Load case	Bending	Weight	Torsion	Weight
Bending	√	1.0		1.0
Torsion		1.0	√	1.0

5

In the window, click .

6

In the window for , type Yaw Sensitivity in the text field.

7

Locate the section. Clear the check box.

1

In the toolbar, click

.

2

In the window for , locate the section.

3

In the table, enter the following settings:


Expression	Description	Evaluate for
comp1.truss.avgr1.thZ	Yaw angle	Stationary

4

In the toolbar, click

.

Yaw Sensitivity

1

In the window, right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

In the text field, type Yaw Sensitivity.

1

In the window, expand the node, then click .

2

In the window for , locate the section.

3

From the list, choose .

1

In the window, click .

2

In the window for , locate the section.

3

From the list, choose .

Yaw Sensitivity

1

In the window, click .

2

In the toolbar, click

.

The tower does not yaw in response to bending, but it is possible to achieve such an effect by stiffening and weakening some of the diagonal bars.

As one would expect, the diagonal bars (at all heights) are the ones that need higher stiffness to reduce yaw in response to torsion.