# Short Column (Uncertainty Quantification)

This page shows how to perform Uncertainty Quantification of the Short Column with COSSAN-X.

## Contents

# Problem Definition in COSSAN-X

## Define a Project

Before the model can be analysed it must first be defined. In the first step create a new project named 'ShortColumn': This can be done by pressing the 'new' icon, or from the menu File->New->Project.

## Input

### Parameters

In the next step, the constant input parameters, i.e. the width **b** and the depth **h** are specified. Clicking with the right button at the folder 'parameter' and selecting 'Add parameter' the following wizard appear on the screen (see also Parameter (wizard)):

The input values are specified as shown below and the process is repeated for h

The process is saved by pressing Ctrl S

### Random Variables

Next a probability distribution is associated to each uncertain quantity as described in the model table above. This is done by right clicking on the sub-folder *Random Variables* and selecting *Add Random Variable* which prompts an input mask to pop up in which the uncertain quantity is described. For the random variable yield stress, the distribution is set to lognormal, the mean to 5 and standard deviation of 0.5 is inputed

After the random variable is computed and saved, the CDF and PDF can be seen in the chart next to its inputs. This process is repeated for all the random variables

**Random variable sets**

Following that, the random variables are specified. Random variable sets are utilised as a manner of defining correlation between random variables. Random variable sets are created by. clicking the sub-folder "Random Variable Sets" and selecting "Add Random Variables Set"

Once the finish button is pressed, the editor for the random variable set will open as shown below. The GUI supports "drag and drop". therefore, the list is filled by just dragging all icons in the workspace to the window on the right hand side.

All random variable in the set are assumed to be uncorrelated. The bending moment is strongly correlated with the axial force. In this case, a correlation coefficient of 0.5 has been selected. Note that zero correlations can be left blank.

This completes the input in the sub-folder "Random Variable Sets".

**Functions**

Similar to generating parameters and random variables, each function is generated by right clicking the sub-folder "Function". Doing so pops up a window which allows the user to create a new function where the user provides the function with a name and description. We use the name "f1"

Once again, when the finish button is pressed, the editor window pops up for the function and it is defined as follows below

## Evaluator

Evaluators are used to define a model. They are low-level definitions of the functional relationship between the inputs and outputs of the model.

### Matlab Script

The use of the Matlab script in calculating the values of interest. Creating a Matlab script is done following the following path. The following figure also demonstrates it visually

Evaluators => Matlab Files => Right click => Add Matlab Script

Carrying on it is important to specify the input factors needed to calculate the value of interest. This is done by pressing the plus button of the Matlab script in order to add all the required fields. Following that, a name must be given to the calculated value of the function as shown in the following figure

In order to edit the Matlab script, the "script/function" toggle is selected. The Matlab script is as shown below.

**Performance Function**

The performance function is used to define the domain of the model. The domain is defined into two sets, the safe set and the failure set. For more information see Performance function definition.

## Model

Defining the model is one of the most important steps in investigating the effects of uncertainties. "Physical Model" allows the creation of independent samples by various methods to study the variability (scatter) of the response. "Optimisation Models" is selected when the response should be optimised in same sense. "Probabilistic Models" is related to reliability analysis.

The first step in creating a physical model is selecting the "physical model" with the cursor clicking the right mouse button and selecting "Add Physical Model". Following that, a new window pops up demonstrating the names which can be edited. Default values are selected by clicking the finish button

Once again, the editor window appears. the evaluators and random variables can be added at this stage. If there is a red message at the top of the page, it is indicating that there is not enough information to complete a successful simulation analysis.

Clicking the + sign of the evaluator list allows a new window to pop up. This window shows the available options that can be added to the list. In this case, Eval is selected and then the ok button is pressed.

Next, the inputs and outputs related to the evaluator are revealed as shown in the following:

## Uncertainty Quantification Analysis

The simulation can be started by clicking the small white triangle within a green circle at top right as shown

## Running the Analysis

The next window allows the user select the desired type of analysis from the following:' Design of Experiments', 'Sensitivity Analysis', 'Uncertainty Quantification' and 'Userdefined Analysis'. In this tutorial 'Uncertainty Quantification' is selected.

after clicking 'Next' as shown in the previous window, various options for simulation are provided. Monte Carlo sampling procedure is chosen. At the right, the number of independent samples can be declared. It is possible to split the computation into several batches which might be processed by different computers using parallel computing.

In this case, only a single batch is used. Selecting a zero maximum runtime means there is no limit for the runtime.

After pressing next again, the grid settings could be specified (how the analysis is performed). In this case, default settings are used

The calculations commence once the finish button is clicked. A message of successful completion appears when simulation is finished.

## Show the Results

Results are accessed via the "Analysis" section. The results are stored in sub-folders according to the performed analysis. In this case, Uncertainty Quantification has been performed.

**Uncertainty Quantification**

The results folder of the uncertainty quantification is expanded in order to reveal all the involved variables defined in the input. The results are facilitated visually by selecting the output view. Alternatively, the 'Output view' can also be selected from the menu 'Window'>'Open Perspective'>'Output'.

**Scatter plots**

The following illustrates the 'Scatter View' and the 'Parallel Coordinates' view:

To illustrate all 1000 realisations, the variables can be chosen using the mouse and dragged to the scatter view

**Parallel coordinates**

The 'Parallel Coordinates View' allows to the user to reveal the correlations between several variables as shown:

**Table view**

A new table view is opened by selecting the icon on the toolbar and dragging and dropping the quantity of interest on the table

**Histogram**

A new table view is opened by selecting the icon on the toolbar and dragging and dropping the quantity of interest on the histogram view