# Stochastic Approach

In real life, the values of the product design variables, like geometry, material properties, load magnitudes etc., always contain some uncertainty. Solving engineering problems with uncertainties requires a stochastic approach.

## What is Stochastic Approach?

Applying a stochastic approach in product design allows to:

• Study product behavior in close to real life conditions
• Decrease the effect of uncertainties
• Increase product reliability and safety
• Validate product robustness under various conditions

Design variables

Stochastic variables

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Stochastic responses

Applying a stochastic approach to product design manually is not an easy task and is insufficient in most of the cases. pSeven provides a variety of tools to deal with uncertainties and to find robust designs in an automatic manner:

• Uncertainty Quantification (UQ)
• Robust Design Optimization (RDO)
• Reliability-Based Design Optimization (RBDO)

## Uncertainty Quantification (UQ)

Specialists from a wide spectrum of industries face the need to evaluate the influence of uncertain parameters of a product, like material properties or operating conditions, on its technical and operational characteristics. Uncertainty Quantification (UQ) in pSeven addresses this need and allows engineers to significantly improve quality and reliability of designed products and manage potential risks at early design, manufacturing and operating stages.

UQ is used to assess model design point taking into account all possible deviations of input parameters and their influence on the output. Uncertainties of the input parameters are described with distributions, based on experimental data, production constraints, best practices or engineering judgment. The most important part of UQ process is defining the model assessment criteria, for example, failure conditions. As a result of UQ user obtains a distribution of these criteria, including mean and dispersion values which allow to evaluate the model reliability and make better engineering decisions.

### Uncertainty Propagation

pSeven allows efficiently quantify and deal with uncertainties in design variables and responses:

• Manual selection of input distribution types
• Automatic fitting of input and output samples to the available distribution type
• Creating non-parametric distributions
• Running Sensitivity Analysis to estimate the influence of uncertainties on product behavior

Distributions of uncertain design variables

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Model

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Estimate the influence of uncertainties

### Reliability Analysis

Some realizations of the model with uncertainties may satisfy all the design criteria, but some may fail. pSeven allows to easily assess the reliability of the product:

• Variety of methods for model sampling available (eg. Monte Carlo, LHS etc.)
• Approximation is used to drastically reduce the number of heavy simulation runs
• Estimation of failure probability (N failures / N of simulations)

Distributions of uncertain design variables

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Model

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Introduce constraint, e.g. stress safety factor

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Failure domain found

## Robust Design Optimization (RDO)

Sometimes product behavior must be stable, but the best possible solution may be not robust to slight variations. Robust Design Optimization is a search for good and robust solutions rather than the best possible solutions.

Single- and multi-objective robust optimization in pSeven supports virtually all possible formulations, including probabilistic and quantile type constraints.

Non-robust response

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Robust response

It is based on the state-of-the-art stochastic approach that supports both direct and surrogate-based optimization, so it is suitable for expensive to evaluate problems. The basis of this approach is a careful adjustment of the distribution realizations number of uncertain parameters. Only a small number of random realizations away from optimal solution needs to be considered to change the uncertain parameter itself. When the optimal solution is approached, the number of distribution realizations will be increased.

The unique feature of this approach is that it provides both the solution and the corresponding uncertainty estimations of objective and constraints values. Another advantage of robust optimization in pSeven for engineering applications is that explicit distribution law of the uncertain parameters is not required, it is enough to provide their distribution empirically.

## Reliability-Based Design Optimization (RBDO)

Sometimes, underperformance or even failure of the product due to uncertainties are impossible to avoid completely or acceptable with some probability, e.g. for temporary or non-critical elements.

Reliability-Based Design Optimization is a search for solutions with design constraints that will be satisfied with a given probability (also called chance constraint).

Deterministic response

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Reliable response

pSeven allows combining all the existing optimization functionality with Reliability-Based approach, including:

• Statistics analysis that estimates which design point is better with a limited number of solver runs
• Surrogate-Based Optimization to reduce the number of expensive model evaluations
• Multi-objectiveness

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