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What isUncertainty Quantification (UQ)?

What is Uncertainty Quantification (UQ)?

What is Uncertainty Quantification (UQ)?

Definition of Uncertainty Quantification:Uncertainty Quantification (UQ) is the modelling and evaluation of the impact of imperfectlyknown information on the design and development of products or processes to supportdecision making.

Why do we need UQ?Recent years have seen a drive to replace reserve margins-based deterministic designprocedures with quantified variation-based design procedures using stochastic analysis eitherto optimize design solutions further, or to quantify design robustness and reliability moreprecisely. With the traditional approach of deterministic design, all uncertainties are assumedto be accounted for with reserve margins, often in combination with a "worst case scenario"approach to the values used in deterministic calculations. As the complexity of designsolutions increases; however, it becomes more and more difficult to define with confidencewhat the absolute "worst case scenario” might be, and to identify the appropriate reservemargin for any given situation. This results in situations where the designs are over-designedto an unknown degree, or even potentially under-designed. The use of UQ techniquesaddresses these issues directly, as shown in Figure 1.

UQ also brings benefits in validation of simulation against physical test, including theuncertainties associated with both the physical test data and the simulation results in acomparison that allows a confidence level to be associated between the two. This approachallows for a better understanding of the true situation in comparison with a simpleobservation between two sets of numbers. Thus, the stated confidence provided by UQsupports more reliable and consistent decision-making.

UQ can also be used in the model development process to quantify the contribution of modelassumptions on the predicted response. Performing sensitivity analyses on the assumptionsidentifies opportunities for model improvements or the need for additional experimental datato reduce the uncertainty around the assumptions. Figure 2 shows a scenario of propagating

Figure 1: Difference between design rule understanding and UQ

variations in inputs through the model to predict the reliability function and probabilisticsensitivity factors, with the dotted lines representing uncertainty in the assumed inputs andhence establishing the bounds on the predicted reliability. The vertical red lines representtraditional conservative values on inputs and may result in non-optimal designs. The verticalblack lines represent nominal or mean values.

Simulation methods and tools are used in combination with physical testing to perform thestochastic assessments, as it would not be possible to generate all of this information usingphysical testing alone. One of the key steps for a stochastic analysis is the formulation of thestatistical models to characterize imperfect and/or unknown information. These models arerequired to allow the stochastics methods to sample the inputs and build a picture of thevariation in the key outputs. The following section outlines the steps involved in developingstatistical models.

Key Steps in the UQ process:1. Identify all inputs that are influencing the critical simulation outputs.

2. Collect all available data and estimations about the uncertainties such as: availablemeasured data, available upper and lower limits as a result of quality processes in place, orestimations from relevant experts, or other valid sources.

3. Classify your input data into irreducible and reducible uncertainties, as these will have tobe treated differently in any stochastic assessment [Reference 1]. An irreducible uncertainty(aleatory uncertainty) is an inherent variation associated with the physical system beingmodeled which is characterized by a probability distributions. Reducible uncertainty(epistemic uncertainty) is the lack of precision in a measured or calculated value that canbe reduced by gathering more data, observations, or information. Reducible uncertainty istypically covered by conservative assumptions. It is also called "lack of knowledgeuncertainty". In real life applications, both kinds of uncertainties are present. For aleatory

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Figure 2: Uncertainty propagation process.

What is Uncertainty Quantification (UQ)?

uncertainties, traditional statistical methods can be used to analyze the data and derivethe appropriate statistical models. However, since the quantity and quality of available datafor the inputs will determine the type of statistical model that can be developed, it isimportant to understand the influence on these statistical models, and to decide how thiswill be treated in any subsequent UQ analysis. For epistemic uncertainties, Bayesianapproaches and the principle of maximum entropy can help turn information and expertopinion into a distribution. Some examples for common sorts of information are given in[Reference 2].

4. The next step for any UQ is to perform a sensitivity analysis of important responses to allof the uncertainties. The purpose is to check any assumptions about parameterunimportance, linearity, or dimensionality of the UQ analysis [Reference 3].

5. After developing a good understanding of the important inputs, the next step is topropagate all this variability to the output side of the simulation or experiment tocharacterize the variability of the key outputs. A multitude of methods is available toperform this propagation. Some generally available methods are illustrated in Figure 3. Themost general and accurate method to approximate response variability and relatedprobability is the Monte Carlo Sampling technique. However, because of the very largenumber of samples needed to produce precise results, this method is not suitable forsituations where the calculations will be too costly. In these cases, alternative methods canbe used, which continue to be very active research topics.

6. Comparing the estimated variation in the key outputs (from the previous step) with thedefined allowable limits and design targets allows for a much better decision-makingprocess during the design and development of (engineering) systems.

Figure 3: Some methods for uncertainty propagation.

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Summary:This short flyer defines the term uncertainty quantification and summarizes the key steps forany UQ process. These steps are agnostic to the application area. If you are interested in thisarea and want to help shape future guidance materials please contact NAFEMS to become amember of the Stochastics Working Group (stochastics@nafems.org).

References:[1] The America Society of Mechanical Engineers. 2018. Codes & Standards - V&V VERIFICATION ANDVALIDATION IN COMPUTATIONAL MODELING AND SIMULATION. [ONLINE] Available at:https://goo.gl/dKs5ys (Accessed 6 July 2018)

[2] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML, “Evaluation of measurement data - Supplement 1to the Guide to the expression of uncertainty in measurement - Propagation of distributions using aMonte Carlo method”, Joint Committee for Guides in Metrology, JCGM 101:2008, 2008,https://goo.gl/382nXp (Accessed 5 July 2018)

[3] Bartholomew. P, “What is Sensitivity Analysis”, NAFEMS, 2014https://goo.gl/Y1cXiK (Accessed July 2018)

Simple Example: Problem description:A design team is asked to create a coat hook that differs from the classical shape. They decideto shape it as a cantilever beam with a small outstanding rim to prevent anything placed onthe hook from sliding off. Anticipating a deflection they realize that the coat, or whatever isplaced on the hook, will slide to the rim and therefore the design can be treated as a simplecantilever beam with a tip load, see Figure 4.

Since the design team are uncertain with respect to the application of the hook, they ask formore information. The response is that the customer does not yet know where the hook willbe used. It could be used for a kindergarten (one small jacket on one hook) or for a theatre (acouple delivers two winter coats at the cloakroom). The answer is very vague, but it isabsolutely certain that the load will be more than children’s outerwear on a hook. As aguideline, they use a mass of 15 kg. The design team struggles with the vague response ofthe client in respect to the anticipated loading. They have a strong feeling that the client isjust guessing and the uncertainty, caused by a lack of knowledge, can be reduced.

Figure 4: Cantilever beam with tip load

What is Uncertainty Quantification (UQ)?

The determinstic approach Based on their experience the design team focuses on a cantilever rod with a circular crosssection and a tip load, F ≈ 150 N. They decide to use a length, L = 70 mm. They have hadexperience working with austenitic steel and select AISI316 with a yield stress, s yeild = 225MPa. They decide that an allowable design stress limit of 60% of the material yield stress isacceptable for the operational loading.

These inputs define the circular cross section of the rod to be:

The uncertainty involved

The design team is very confident regarding the length of the rod, since this is the only aspectover which they have absolute control. Nevertheless they realize that even in this case thelength of 70 mm is a mean value with a standard deviation. They are convinced that they canreach a standard deviation of 1.0 mm.

The rod supplier offers diameters in whole millimetres and looking at the ample margin to theyield stress they decide to use a rod diameter of 9.0 mm. These diameters are mean valueswith a standard deviation of 0.45 mm. The production process of the rods has amanufacturing end control and diameters in excess of 9.4 mm are removed as well asdiameters less than 8.5 mm. The rolling process delivers diameters that possess a normaldistribution, but due to removal of excessive diameters the distribution is truncated.

The yield stress is a minimum specified yield stress which implies that the mean value is higher.Via the supplier they hear from the steel mill that the material has a mean yield stress value of270 MPa, a log-normal distribution and a standard deviation of 14 MPa.

The uncertainties mentioned so far are irreducible and called aleatoric. The tip load has notbeen quantified in detail but based on the client’s information they assume a maximum loadof two coats, large size, wet from stormy weather with the possibility that the pockets containa substantial purchase from a hardware store. As a minimum load a small child coat in akindergarten is assumed. This information is translated into a uniform distribution between 90N and 210 N. This is called an epistemic uncertainty as this can be easily reduced by gatheringmore information from the use case of the coat hook.

An overview of the input parameters is presented in Table 1.

Table 1 Statistical input

Variable Mean value Standard deviation Distribution type

Length [mm] 70 1.0 Normal

Diameter [mm] 8.5 < 9.0 < 9.4 0.45 Truncated normal

Yield stress [MPa] 270 14 Log-normal

Tip load [N] n/a n/a Uniform (90 to 210)

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UQThe probability of failure is calculated, i.e. the probability that the bending stress, S exceedsthe yield stress, Y or Pr (Y-S < 0). Figure 5 shows the histogram of the difference between thebending stress and the yield stress, retrieved by the response surface method, followed byMonte Carlo Sampling to obtain the required data. The analysis is performed and theprobability that the difference between the bending stress and yield stress is smaller than zeroamounts to 9.4x10-5 or 1 in 10638 failures. The sensitivity to the input parameters isillustrated in Figure 6, where the biggest influence originates from the tip load. Since thisinput contains a large variance it dominates the response. The result is less sensitive to thelength, since the variance is low.

The designer of the coat hook is satisfied with the result and proposes a diameter of 9.0 mm.However, since the deterministic approach resulted in a diameter of 9.3 mm the client mustbe convinced that the proposed design with 9.0 mm is acceptable. In this discussion thestochastic analysis with the quantification of the failure probability provided the necessaryinformation to convince the client.

Figure 5: Histogram

Figure 6: Sensitivity analysis

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