## Statistical hot spot detection and visualization

Critical locations cannot always be clearly determined on the basis of a single simulation. This is particularly the case when the position of the maximum jumps or slowly changes its position. In this case, a statistical analysis can help. Ansys SoS provides a variety of statistical functions whose results can be displayed in 3D.:

- Encapsulating (lower and upper) value limits
- Mean, standard deviation, variance
- Linear correlations and CoD with respect to input parameters
- Nonlinear sensitivity indices with respect to input parameters (F-CoP)
- quantile values, k*sigma values and exceedance probabilities for fixed limit values
- Cp, Cpk statistics
- Standard error for mean value and variance
- Frequency of cracks

You can use a script to make your own evaluations.

Practical auxiliary functions allow the simple selection of extreme points with regard to certain robustness measures. The selected points can be extracted for all DOE designs for further processing in Ansys optiSLang or Excel.

Different visualization modes and various configuration options round off post-processing. You can choose between different representations. Most pallets are designed for users with red-green blindness. The interior of 3D structures can also be made visible with semi-transparent isosurfaces or through sectional planes.

## Data-based reduced-order modeling (ROM): sensitivity analysis and approximation with FMOP

ROMs are of great importance for system simulation and for digital twins. In order to optimize the maintenance and operation of a system, a detailed product simulation must be linked to sensor data for an accurate prediction of the characteristic values (e.g., the service life of turbine blades). The requirements for the response time of digital twins can only be met if the detailed simulation models are simplified. The classical "physics-based" approach of ROMs uses a matrix condensation whose formula still contains the physical influences of the input variation on the response variables. However, this type of reduction is often only suitable for linear systems. A data-based ROM is the better alternative for simulating nonlinear systems. The method uses function models for approximating the response surfaces, taking into account the influence of input variations on the response variation depending on the given parameter set. For the field data of the input or response variables, SoS uses the field metamodel of optimal prognosis (FMOP) to approximate signals, FEM solutions or geometric deviations.

SoS offers the possibility to visualize FMOPs interactively via slider. The evaluation of an FMOP can be binary coupled with a DLL, so that the metamodels can be evaluated directly in C++, Python, Matlab or as a background process on a website. Using SoS nodes, the evaluation of an FMOP can also be carried out in an optimization in optiSLang. The field prediction measure F-CoP provides information showing at which points of the FEM network the FMOP has a high or low prediction quality.

The quality of the FMOP can also be used to identify global sensitivity measures for the input parameters. Therefore, the sensitivity of the input parameters can be assigned to the different positions on the FEM mesh.

## Random fields based on measurements

Often one is faced with the question of how to describe a field quantity (e.g., geometry variation, stress, strain variation, load strain curve). For example, the traditional way would be to describe signal data by parameters.

If enough measurements or simulation results from a DOE are available, parameterization can be performed using empirical random fields. The dominant variation patterns are automatically identified from the data. The variation patterns are sorted according to their importance. A parameter is then assigned to each variation pattern.

This auto-parameterization is optimal, i.e., it is ensured that a maximum of variation can be represented with the smallest possible number of parameters. The parameterization can be used to:

- Find a statistical description (correlation in space/time, distribution type, etc.) based on measurements and store it for subsequent applications
- Generate new random designs (e.g., signals, geometries, etc.)

An integration in optiSLang is available for generating random fields.

## Random fields based on autocorrelation models

If there are no or not enough measurements available, field variations can be depicted using autocorrelation models. These models are based on the assumption that points being spatially close to each other are strongly correlated (i.e., may even have similar values), and points that are farther away are less or not correlated.

This can be represented by autocorrelation functions, which depend on the distance between two points and calculate the correlation. The degree of distance dependence can be controlled by a parameter (correlation length).

In SoS, these models are implemented as parametric models. There are two variants:

**Synthetic random field model**

An autocorrelation function is determined for each point on the network and then a minimum number of parameters is determined by Karhunen-Loeve decomposition to obtain predetermined statistical properties (mean, standard deviation, correlation length).

**Free-form variation model**

Here a range of points is selected in advance (or is determined automatically) and, for these points, an autocorrelation function (interpolation) is then directly linked to a scaling parameter.

## SoS Plugin for Ansys Workbench

Statistics on Structures offers a plugin for Ansys Mechanical. You can use it:

- To create geometric variations with a synthetic random field model
- To create geometric variations with a free-form variation model
- To export any result data to SoS

For the creation of geometric variations, the plugin can access the mesh and named selections from Ansys. You can define the statistical parameters via the Ansys GUI. The visualization of Ansys Mechanical is used to display the scatter shapes or the resulting geometry change as a color plot. The resulting geometry can be displayed via the RST file in Ansys Mechanical. The parameters of the random field models can either be modified manually or made available via the Ansys Workbench parameter set in optiSLang for variation purposes.

The data export serves to prepare the mesh and the data for parts of the structure in a format suitable for SoS. The data can then be very easily imported into SoS and evaluated for statistical analysis or for the creation of an FMOP.