COMSOL 6.4 - Uncertainty Quantification of a Bracket

Uncertainty Quantification of a Bracket — Fillet Version

This example demonstrates how to use the Uncertainty Quantification Module by running a series of uncertainty quantification studies for a steel bracket.

This type of bracket can be used to install an actuator that is mounted on a pin placed between the two holes in the bracket arms. The design objective is that the actuator’s horizontal misalignment, φ, should not be too large.

Figure 1: Bracket geometry with the quantity of interest, the misalignment angle, φ, indicated.

The geometry is created using the 3D fillet functionality that is available in the Design Module. If you do not have access to the Design Module, a version without 3D fillets is also provided; see Uncertainty Quantification of a Bracket. A few of the model parameters are slightly changed for the version without fillets but otherwise the instructions are identical.

Model Definition

The geometry is fully parameterized with parameters according to the table shown in Figure 2.

Figure 2: Geometry parameters.

The mesh is customized to have about two elements across the thickness of the material.

Figure 3: Meshed geometry.

In this analysis, the mounting bolts are assumed to be fixed and securely bonded to the bracket. One of the arms is loaded upward and the other downward. The loads are applied as a pressure on the inner surfaces of the holes, and their intensity is P0cos(α), where α is the angle from the direction of the load resultants.

Figure 4: Applied loads.

This force is assumed not to vary in the uncertainty quantification studies. Similarly, the material properties for a generic structural steel are assumed not to vary.

The misalignment angle is chosen as the quantity of interest (QoI). The design objective is that the actuator’s horizontal misalignment angle φ does not exceed 0.1 degree. The angle is defined as a variable; for details, see the Modeling Instructions section.

The parameters to be varied are the geometric dimensions of the bracket.

The parameters that participate in the uncertainty quantification analysis are all assumed to be normally distributed around their nominal values according to the table in Figure 5, which corresponds to the screening and sensitivity studies, as described later.

Figure 5: Input-parameter distributions used in screening and sensitivity studies.

The Uncertainty Quantification Studies

The Uncertainty Quantification Module provides four different study types:

•

Screening, MOAT

Identifies the most influential inputs, for each QoI

Is based on the Morris One-At-a-Time (MOAT) method

Outputs MOAT mean and MOAT standard deviation values

•

Sensitivity Analysis

Computes the fraction of impact for the inputs, for each QoI

Outputs first-order and total Sobol indices

•

Uncertainty Propagation

Computes the statistical variation of the QoI

Outputs a kernel density estimation (KDE) plot representing an estimate of the probability distribution of the QoI

•

Reliability Analysis

Computes the probability for the fulfillment of a condition based on the QoI

For example, what is the probability P(φ > 0.1)

•

Inverse Uncertainty Quantification

Uses experimental data to backtrack and uncover the statistical properties of certain input parameters, starting with some initial assumptions about them (prior probability distribution).

Applies surrogate models and statistical methods to refine the characterization of these inputs, providing (posterior) probability distributions, as well as a confidence interval table containing the mean, standard deviation, min/max values, and confidence bounds.

This example illustrates the first four of these study types.

For more information, see the Uncertainty Quantification Module User’s Guide.

Surrogate Models

To get statistical data based on a physics model you need to run a lot of simulations, varying the parameters of the inputs according to their probability distributions. For a 3D model, this might be computationally unfeasible. To get around this problem, the Uncertainty Quantification Module first builds up a so-called surrogate model that is used for sensitivity analysis, uncertainty propagation, and reliability analysis (but not for screening).

This process is typically adaptive and the surrogate model can approximate the original model to a high degree of accuracy (which can be modified by the user). The Uncertainty Quantification Module uses two different types of surrogate models:

•

Sparse Polynomial Chaos Expansion (SPCE)

This surrogate model improves its accuracy by adaptively solving the full model and thereby adding new QoI data by using sequential Latin hypercube sampling.

•

Gaussian Process (GP)

This surrogate model improves its accuracy by, using information from the current Gaussian Process surrogate model, adaptively solve the full model for new carefully selected sets of parameter values.

Results and Discussion

The uncertainty quantification study gives the kernel density estimation plot shown in Figure 6 with associated confidence interval information in the QOI confidence interval table, as shown in Figure 7.

Figure 6: Kernel density estimation.

Figure 7: QoI confidence interval.

The reliability analysis study shows that the probability for the misalignment angle φ not to exceed 0.1 degree is about 0.003 or 0.3%. This result is made available in the table, as shown in Figure 8.

Figure 8: Probability for conditions.

Note that this value is sensitive to minute changes in the mesh and may vary slightly depending on the COMSOL Multiphysics version you are running.

It is now up to the designer to decide whether this is an acceptable level or risk or if a redesign and/or further studies are needed.

To be more conservative about the uncertainty propagation and reliability analysis, we can choose to include all parameters in the analysis. Running the computationally much more demanding uncertainty propagation and reliability analysis for all parameters (not demonstrated here) shows that the probability for the misalignment angle φ not to exceed 0.1 degree is about 0.8%.

Uncertainty_Quantification_Module/Tutorials/bracket_uncertainty_quantification_fillet

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > .

Click .

Click

In the tree, select > .

Click

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose . This is necessary because the prepared geometry sequence contains a feature.

In the toolbar, click and choose .

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

In the toolbar, click

Global Definitions

The geometry sequence already provided a set of geometry dimension parameters. Now define this additional set of parameters for the boundary load and mesh density.

Parameters 2

In the toolbar, click

and choose > .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
P0	2.5[MPa]	2.5E6 Pa	Force per area
dx	lp+ts	0.198 m	X-distance between pin holes
hmax	ts/1.5	0.0053333 m	Max element size
hmin	ts/2.5	0.0032 m	Min element size
YC	-ls+r2	-0.3 m	Y-coordinate of hole center

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Definitions

Next, define a series of selections to be used for the constraints and loads.

Bolt Holes

In the window, expand the > node.

Right-click and choose > .

In the window for , locate the section.

From the list, choose .

Select the checkbox. This setting greatly facilitates the selection of curved surfaces. In this case use it to select the four bolt holes. It is sufficient to select one boundary per hole; the three other boundaries will be selected automatically.

In the text field, type Bolt Holes.

Left Pin Hole

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select the checkbox.

Select Boundaries 4, 5, 7, and 8 only.

In the text field, type Left Pin Hole.

Right Pin Hole

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select the checkbox.

Select Boundaries 52–55 only.

In the text field, type Right Pin Hole.

Pin Holes

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Locate the section. Under , click

In the dialog, in the list, choose and .

Click .

In the window for , type Pin Holes in the text field.

Define average operators that will be used for computing the average z-directional displacement of the pin holes.

Average 1 (aveop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Average 2 (aveop2)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Variables 1

Right-click and choose .

Define the quantity of interest (QoI), the misalignment angle, as well as auxiliary variables for the pin hole z-coordinates.

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
z1	aveop1(w)	m	Left pin hole center
z2	aveop2(w)	m	Right pin hole center
phi	atan((z2-z1)/dx)[1/deg]		Misalignment angle

The output from the atan() function has the unit radian, which is unitless. The conversion factor [1/deg] corresponds to multiplication by 180/pi for converting to degrees.

Now, define an analytic function load used to apply the spatially varying load on the pin holes.

Analytic 1 (an1)

In the toolbar, click

In the window for , type load in the text field.

Locate the section. In the text field, type F*cos(atan2(py,abs(px))).

In the text field, type F, py, px.

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

In the table, enter the following settings:


Argument	Unit
F	Pa
py	m
px	m

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Ensure that the mesh density is fine enough to resolve all the stress gradients reasonably well.

Size

In the window, under > click .

In the window for , locate the section.

Click the button.

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

In the text field, type hmin.

Click

Solid Mechanics (solid)

Apply fixed constraint boundary conditions on the bolt holes.

Fixed Constraint 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Next, apply the spatially varying load.

Boundary Load 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Specify the fA vector as


0	t1
0	t2
load(-P0,Y-YC,Z)(sign(X)Z>0)	n

Locate the section. From the list, choose .

Study 1, Static

First, run a static analysis with no uncertainty quantification analysis.

In the window, click .

In the window for , type Study 1, Static in the text field.

In the toolbar, click

Results

Volume 1

Now, let us assume that we are only interested in values up to 100 MPa.

In the window, expand the node, then click .

In the window for , click to expand the section.

Locate the section. From the list, choose .

In the toolbar, click

Locate the section. Select the checkbox.

In the text field, type 0.

In the text field, type 100.

In the toolbar, click

Study 1, Static

In the window, right-click and choose > .

Study 2, Screening

In the window for , type Study 2, Screening in the text field.

Next, add a screening analysis to see which input parameters are most significantly impacting the misalignment angle (QoI). The screening study is added as a study reference which means it refers back to the already defined static study.

The parameters that participate in the uncertainty quantification are all assumed to be normally distributed around their nominal values, according to the instructions below. The mean and standard deviation, as well as the max and min limits are all defined in terms of their nominal parameters (from > ).

Uncertainty Quantification

In the window, under click .

In the window for , locate the section.

Click

In the table, enter the following settings:


Expression	Include study-dependent input
comp1.phi	Reduce to single global output

Locate the section. Click

eight times.

In the table, enter the following settings:


Parameter	Source type	Source description
ts (Material thickness)	Analytic	Uniform
lp (Cross-plate length)	Analytic	Uniform
ls (Side length)	Analytic	Uniform
hm (Side height)	Analytic	Uniform
wp (Cross-plate width)	Analytic	Uniform
wf (Flange width)	Analytic	Uniform
fr1 (Fillet radius)	Analytic	Uniform
r1 (Pin-hole radius)	Analytic	Uniform

In the table, click to select the cell at row number 1 and column number 1.

From the list, choose μσ.

In the text field, type ts.

In the text field, type 0.01*ts.

From the list, choose .

In the text field, type 0.99*ts.

In the text field, type 1.01*ts.

In the text field, type m.

In the table, click to select the cell at row number 2 and column number 1.

From the list, choose μσ.

In the text field, type lp.

In the text field, type 0.01*lp.

From the list, choose .

In the text field, type 0.95*lp.

In the text field, type 1.05*lp.

In the text field, type m.

In the table, click to select the cell at row number 3 and column number 1.

From the list, choose μσ.

In the text field, type ls.

In the text field, type 0.01*ls.

From the list, choose .

In the text field, type 0.95*ls.

In the text field, type 1.05*ls.

In the text field, type m.

In the table, click to select the cell at row number 4 and column number 1.

From the list, choose μσ.

In the text field, type hm.

In the text field, type 0.01*hm.

From the list, choose .

In the text field, type 0.95*hm.

In the text field, type 1.05*hm.

In the text field, type m.

In the table, click to select the cell at row number 5 and column number 1.

From the list, choose μσ.

In the text field, type wp.

In the text field, type 0.01*wp.

From the list, choose .

In the text field, type 0.95*wp.

In the text field, type 1.05*wp.

In the text field, type m.

In the table, click to select the cell at row number 6 and column number 1.

From the list, choose μσ.

In the text field, type wf.

In the text field, type 0.01*wf.

From the list, choose .

In the text field, type 0.95*wf.

In the text field, type 1.05*wf.

In the text field, type m.

In the table, click to select the cell at row number 7 and column number 1.

From the list, choose μσ.

In the text field, type fr1.

In the text field, type 0.01*fr1.

From the list, choose .

In the text field, type 0.95*fr1.

In the text field, type 1.05*fr1.

In the text field, type m.

In the table, click to select the cell at row number 8 and column number 1.

From the list, choose μσ.

In the text field, type r1.

In the text field, type 0.01*r1.

From the list, choose .

In the text field, type 0.95*r1.

In the text field, type 1.05*r1.

In the text field, type m.

In the toolbar, click

Results

MOAT, comp1.phi

Click the

button in the toolbar.

The screening results indicate that the Side length (ls) and the Cross plate width (wp) parameters are the most influential on the quantity of interest. However, several of the other parameters also appear to be significant. A high value of the MOAT mean means that the parameter is significantly influencing the quantity of interest. A high value of the MOAT standard deviation means that the parameter is influential and that it is either interacting with other parameters and/or that it has a nonlinear influence.

Study 2, Screening

The next step is a sensitivity analysis. Use the results from the screening to decide which parameters to include in the sensitivity analysis. Sensitivity is more computationally demanding than screening and for this reason we would prefer to pick a subset of the parameters used for the screening study. However, in this example, we would like to learn as much as possible about the model and we pick all the available design parameters. We do not need to type all of the uncertainty quantification parameters again but we can define the new Uncertainty Quantification study for the sensitivity analysis by reusing the information in the screening study.

Uncertainty Quantification

Right-click and choose > .

Study 3, Sensitivity

In the window, click .

In the window for , type Study 3, Sensitivity in the text field.

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Results

Sobol Index, QoI1

The sensitivity analysis is based on the Sobol method, also known as variance-based sensitivity analysis. The result of the sensitivity analysis is a set of Sobol indices and an associated Sobol table and Sobol plot. There are two different types of Sobol indices: first-order index and total index. The first-order index of a parameter shows the sensitivity by varying this parameter alone. The total index shows how much a parameter contributes to the overall sensitivity.

In this case, the first and total indices are equal, up to the computed accuracy, for all parameters which indicates very little or no interaction between the parameters. The Sobol plot indicates that the misalignment angle is most sensitive to the parameters Side length (ls) and Cross plate width (wp). This is consistent with the screening results.

For the final two studies, Uncertainty Propagation and Reliability Analysis, we will, for the purpose of faster demonstration, delete all parameters except for the dominant two: ls and wp. However, to get a conservative estimate of the uncertainties we should in principle include all parameters (see earlier comment).

Study 3, Sensitivity

Uncertainty Quantification

Right-click and choose > .

Study 4, Propagation

In the window, click .

In the window for , type Study 4, Propagation in the text field.

In the window, under click .

In the window for , locate the section.

Ctrl-click to select table rows 1, 2, 4, and 6–8. This is most easily done by first clicking in the table’s upper-left corner and then dragging the bottom border of the pop-out table downward until you see all rows at once. The rows to select are those for the parameters ts, lp, hm, wf, fr1, and r1.

Click

Locate the section. From the list, choose . The default is . This will reuse the previously computed uncertainty quantification results in order to speed up the computation. However, we are changing to , which is slower, but gives a higher-fidelity result.

Uncertainty Quantification 3

In the toolbar, click