# Uncertainty Quantification in Finite Element Analysis (FEA)

## Uncertainty Quantification in Finite Element Analysis (FEA)

In the domain of Linear Elasticity, certain problems have known theoretical solutions. For example, thick-walled cylinders subjected to internal pressure are among these problems. However, in cases where known theoretical equations and formulas are not available, such as the transition region between a thick-walled cylinder and a hemispherical dome (like a pressure vessel), numerical simulation techniques such as the Finite Element Method (FEM) are used. The FEM, despite being typically approximate for a given mesh, has the desirable property of converging towards the theoretical solution as the mesh is refined.

To ensure the accuracy of FEM analysis results, it is essential to verify that the input parameters of the model are correct. This is one of the verification stages. However, in many cases, this is not achieved, and precise definitions of parameters such as the problem geometry, material properties, loading, and boundary conditions (BCs), among others, are unavailable. Nevertheless, in many cases, these parameters are known with a certain degree of accuracy or within specific ranges. A common example in this context is the issue of determining whether structural supports should be assumed to be fixed or simply supported.

## The Importance of Uncertainty Quantification in FEM

Uncertainty Quantification (UQ) examines the degree of uncertainty in input data and assesses how this uncertainty might affect the outcomes of analyses and, consequently, the engineering decisions based on them. This process is highly significant in engineering analyses because, by considering existing uncertainties, more realistic results and more precise engineering decisions can be achieved.

## UQ Standards and Tools

Below are some of the standards related to verification, validation, and uncertainty quantification (VVUQ) published by ASME:

• V&V 10–2019: Standard for Verification and Validation in Computational Solid Mechanics
• V&V 10.1–2012: An Illustration of the Concepts of Verification and Validation in Computational Solid Mechanics
• V&V 20–2009: Standard for Verification and Validation in Computational Fluid Dynamics and Heat Transfer
• V&V 40–2018: Assessing Credibility of Computational Modeling through Verification and Validation: Application to Medical Devices

In addition to these standards, there are other guides and resources for UQ. Moreover, SIMULIA Tosca and Isight are designed for optimization and UQ and can be directly linked and analyzed with Abaqus.

Another tool is UQLab a general-purpose framework developed for Uncertainty Quantification (UQ) at ETH Zurich.  UQLab can be linked to an Abaqus model for conducting sensitivity and reliability analyses, particularly in structural applications like the ten-bar truss.

## We Need to Be Sure of the Input Data!

The Finite Element Method is a powerful tool for analyzing complex engineering problems, but the accuracy of the results is heavily dependent on the accuracy of the model inputs. Therefore, uncertainty quantification is a critical component in the numerical analysis process. This process allows FEA engineers to make more confident engineering decisions by understanding and evaluating the impact of uncertainties present in the input data.

## Conclusion

• Verification is performed to determine if the finite element model fits the mathematical description.
• Validation is implemented to determine if the model accurately represents reality.
• Uncertainty Quantification is conducted to determine how variations in numerical and physical parameters affect simulation outcomes.

Ultimately, UQ is a key tool in improving the accuracy and reliability of engineering analyses, helping engineers achieve more precise results and better decisions by considering all aspects related to uncertainty. This process not only improves the quality of numerical analyses but also enhances confidence in the performance of engineering structures and systems.

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