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Category:Uncertainty

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Introduction

Uncertainty Quantification (UQ) is the science of quantitative characterization and reduction of uncertainties in applications. The purpose of UQ is to determine the most likely outcome from models where inputs are not exactly known and help designers to determine the confidence in modeling predictions. In this way, UQ is inherently tied to the concept of Verification and Validation (V&V) of modeling frameworks, and therefore very important within the context of Integrated Computational Materials Engineering (ICME).

Here Verification is defined as the process of determining that a model implementaion accurately represents the developer's conceptual description of the model (i.e. "doing things right"). Validation is defined as the process of determining the degree to which a model is an accurate representation of the real world for the intended uses of the model (i.e. "doing the right thing").

The concept of uncertainty is often confused for the similar concept of error, however there is a difference. Error is the difference in modeling output with respect to real-world experiments that can arise from modeling deficiencies, such as missing temperature dependency when there is a strong experimental correlation or incorrect or inaccurate computational algorithms. That is, the error is most commonly associated with the mathematics of a modeling framework to describe some physical phenomenon. Uncertainty is the potential for error that is due to the lack of knowledge (i.e. modeling based on experimental data that is not exactly known). That is, the uncertainty is related to the underlying physics of experiments and data collection.

A thorough explanation is available from the Stanford University Uncertainty Quantification work group here from Dr. Iaccarino and colleagues.

Sensitivity and Uncertainty Analysis

Qualitatively, uncertainty is the possibility of error in measurement and modeling of physical phenomenon. Uncertainty, in the context of engineering, is the quantification of error in measurements of an experiment. These measurements, when used for modeling, can effect the accuracy and precision of calculations. Performed in conjunction, a sensitivity analysis seeks to describe the sensitivity (in other words, the expected variability) of model output with respect to changes in model parameters. It allows for the ranking of modeling parameters that might dominate the modeling output. It should be noted, however, that sensitivity itself does not directly translate to determining the critical uncertain parameters of a modeling framework. The uncertainty of a given parameter may be small enough such that the uncertainty contributed to modeling output may be smaller than that of a parameter with low sensitivity.

Types of Sensitivity

For modeling frameworks, two types of sensitivity can be defined: local and global sensitivity.

  • Local sensitivities describe the expected change in model output with respect to a change in a parameter based on mathematical derivatives. Generally, perturbation methods are used to evaluate the local derivatives by approximating using forward finite differences, backward finite differences, or central finite differences. If the equations are well-behaved (not highly nonlinear, closed-form, and continuous) and not overly complex, analytical partial derivatives can be obtained by differentiation and solved directly. Local sensitivity analyses are generally used in the model calibration stage to give an idea of calibration precision and accuracy.
  • Global sensitivities describe the expected change of a large modeling framework (e.g. materials modeling using Finite Element Analysis (FEA)) with respect to changes in modeling parameters, much like local sensitivities. However, due to the complex mathematical nature of FEA, being able to track the sensitivity of the model to a given parameter for each element of the analysis can be tedious and time consuming. For these types of analyses, surrogate models capable of approximating FEA output are used for some modeling output metric (e.g. stress/strain in a location of interest) in the design or analysis of some advanced geometry. These methods are inherently tied to the specific problem being solved and do not offer the transportability of local methods. Global methods are usually employed in design and analysis of complicated geometries to provide a basis for reliability analysis.

Sources of Uncertainty

In Uncertainty Quantification (UQ) there are many sources of uncertainty for engineering systems. From the modeling frame of reference, these uncertainties can come from modeling assumptions and simplifications (intrinsic) or from model calibrations using uncertain data (extrinsic). These extrinsic uncertainties can arise from measurement system errors (e.g. bad calibrations) and simplifications of experimental boundary conditions. Figure 1 details the sources of uncertainty for engineering systems.

Fig 1. Intrinsic (internal) and Extrinsic (external) Uncertainty sources for engineering systems from the modeling point of view.

Types of Uncertainty

Fig 2. Random (precision) and Bias (accuracy) Uncertainty related to a distribution of measurements.
Random Uncertainty
  • Often called Repeatability Error, this type of uncertainty derives from variances in repeated measurements and is directly linked to the precision of a measurment system. The more precise the measurement system, the "slimmer" the distribution (Fig 1).
Bias Uncertainty
  • Bias Uncertainty, also called systematic error, derives from the process of using a measurement system and can be introduced due to human faults such as bad measurement system calibration or inconsistent measurement system use. This type of uncertainty can be harder to quantify since not every technician will use a measurement system the same. To minimize this type of error, it is good practice to create a standard operating procedure so that errors will be uniform across all users as well as calibrating the measurement system to a "gold standard". The more accurate (less biased) the measurement system, the closer a sample mean of a set of measurements will be to the true value.

Quantifying Experimental Uncertainties

Confidence intervals are a general method used in quantifying the uncertainty of experimental measurements. Equation 1 shows the confidence interval associated with the normal (Gaussian) probability distribution. For many engineering applications, measurements are usually assumed to adhere to a normal distribution if data is sparse. However, if considerable data is available such that the actual probability distribution can be obtained, it is best to use a confidence interval for its specific distribution.

\overset{-}{x} - t_{\nu,\alpha} \frac{S_x}{\sqrt{N}} \le \mu \le \overset{-}{x} + t_{\nu,\alpha} \frac{S_x}{\sqrt{N}}                     (1)

Where t_{\nu,\alpha} is an estimator from the sample standard normal distribution with degrees of freedom \nu and significance level \alpha , and S_x is the sample standard deviation of N measurements.

Analysis Methods

Propagation of Uncertainty

Often used for closed form, deterministic models, this methodology uses a mathematical treatment based in statistics to propagate the uncertainty of the measured variables (inputs) to the result (output) of a model. A common methodology, the General Uncertainty Analysis, is explained in depth by Coleman et al [1] and is represented by the following equation:


U_r^2 =\sum_{i=1}^{N} \left(\frac{{\delta}r}{{\delta}x_i}\right)^2 U_{x_i}^2                     (2)


Where \left(\frac{{\delta}r}{{\delta}x_i}\right) is the sensitivity of the output function r to the input x_{i} and U_{x_i} is the component uncertainty associated with the input variable x_i . Since the input uncertainty is a constant, the mathematics of the model can either magnify or reduce the contribution of uncertainty from an input. These methods take advantage of linear superposition principles for models that can be approximated with well-behaved Taylor series expansions. For models that display high nonlinearity where the magnitudes of the sensitivities are disperate, different Uncertainty Quantification (UQ) methods may be needed (e.g. Monte Carlo, Latin Hypercube...).

Statistical Methods

When the application of the propagation method is either too tedious (e.g very large model), impossible (e.g. not closed form), or inappropriate (e.g. highly nonlinear) a statistical method is necessary. This method involves automating model inputs to see how the output responds. The most common of these methods is Monte Carlo Random Sampling. Model inputs are randomly selected from statistical distributions (normal, lognormal, weibull, etc...) that are representative of a set of measurements. Generally a very high number of observations are simulated to adequately describe how the input parameters vary with respect to eachother. Using the output data and descriptive statistics, the statistical distribution of the output can be determined with respect to the input distributions. Figure 3 shows the workflow for a Monte Carlo Random Sampling scheme applied to the MultiStage Fatigue (MSF) model.

Fig 3. Monte Carlo Random Sampling Workflow example.

Applications

Microstructure Sensitive MultiStage Fatigue (MSF) Model

Authors: J. D. Bernard, J.B. Jordan, M. Lugo, J.M. Hughes, D.C. Rayborn, M.F. Horstemeyer

Abstract [2]

The objective of this paper is to quantify the microstructurally small fatigue crack growth of an extruded AZ61 magnesium alloy. Fully reversed and interrupted load-controlled tests were conducted on notched specimens that were taken from the material in the longitudinal and transverse orientations with respect to the extrusion direction.In order to measure crack growth,replicas of the notch surface were made using a dual-step silicon-rubber compound at periodic cyclic intervals.By using mic roscopic analysis of the replica surfaces,crack initiation sites from numer ous locations and crack growth rates were deter- mined. Amarked acceleration/deceleration was observed to occur in cracks of smaller length scales due to local microheterogeneities consistent with prior observ ations of small fatigue crack interaction with the native microstructure and texture. Finally,amicrostructure-sensitive multistage fatigue model was employed to estimate the observ ed crack growth behavior and fatigue life with respect to the microstruc- ture with the most notable item being the grain orientation.The crack growt hrate and fatigue life esti- mates are shown to compare well to published findingsfor pure magnesium single crystal atomistic simulations.

For more information on the method of Uncertainty Quantification (UQ) used see MSF Uncertainty.

Microstructure Sensitive Internal State Variable (ISV) Modeling

Author(s): Kiran N. Solanki

Abstract

Fig 2. Experimentation and Simulation Uncertainties

Understanding the effect of material microstructural heterogeneities and the associated mechanical property uncertainties in the design and the maintenance phases of engineering systems are pivotal not only in terms of successful development of reliable, safe, and economical systems but also for the development of a new generation of lightweight designs and predication capability [3] [1]. Engineering systems contain different kinds of uncertainties found in material and component structures, computational models, input variables, and constraints [4] [5]. Potential sources of uncertainty in a system include human errors, manufacturing or processing variations, operating condition variations, inaccurate or insufficient data, assumptions and idealizations, and lack of knowledge [3]. Since engineering materials are complex, hierarchical, and heterogeneous systems, adopting a deterministic approach to engineering designs may be limiting [6] First, microstructure is inherently random at different scales. Second, parameters of a given model are subject to variations associated with variations of material microstructure from specimen to specimen [7]. Furthermore, uncertainty should be associated with model-based predictions for several reasons. Models inevitably incorporate assumptions and approximations that impact the precision and accuracy of predictions. Uncertainty may increase when a model is used near the limits of its intended domain of applicability and when information propagates through a series of models. Experimental data for conditioning or validating approximate (or detailed) models may be sparse and may be affected by measurement errors. Also, uncertainty can be associated with the structural member tolerance differences and morphologies of realized material microstructure due to variations in processing history. Often, it is expensive or impossible to remove and measure these sources of variability, but their impact on model predictions and final system performance can be profound.

Some of the formidable research challenges associated with the integration of advanced (multiscale) computational tools into a computational design framework include uncertainty quantification and the efficient coupling of such material models with nonlinear static and transient dynamic finite element analysis (FEA) of structures, as well as the development of non-deterministic approaches and solution strategies for design optimization under uncertainty [4]. At CAVS, we employ hierarchy of experimental data and material microstructure information to quantify uncertainties through calibration, validation, and verification in the context of materials design and its failure progression mechanisms [8].

Modified Embedded-Atom Method (MEAM)

Abstract - Calibration

Abstract - Sensitivity and Uncertainty Analysis

References

  1. ↑ 1.0 1.1 Coleman, H.W., and Steele, G.W (2010) Experimentation and Uncertainty Analysis for Engineers.
  2. ↑ Bernard, J.D., Jordan, J.B., Lugo,M., Hughes, J.M., Rayborn, D.C., Horstemeyer, M.F., (2013), IJF (http://dx.doi.org/10.1016/j.ijfatigue.2013.02.015).
  3. ↑ 3.0 3.1 Solanki, K.N., (2008), In Mississippi State University (Eds.), Ph.D. Dissertation (http://sun.library.msstate.edu/ETD-db/theses/available/etd-11062008-200935/).
  4. ↑ 4.0 4.1 Solanki, K.N., Acar, E., Rais-Rohani, M., Horstemeyer, M., & Steele, G. (2009). IJDE 1(2) (http://www.inderscience.com/search/index.php?action=record&rec_id=28446/)
  5. ↑ http://catalog.asme.org/Codes/PrintBook/B89732_2007_Guidelines.cfm
  6. ↑ Solanki, K.N., Horstemeyer, M.F., Steele, G.W., Hammi, Y., & Jordon, J.B. (2009), IJSS (http://dx.doi.org/10.1016/j.ijsolstr.2009.09.025).
  7. ↑ Horstemeyer, M.F., Solanki, K.N., and Steele, W.G., (2005). In the proceeding of int. Plasticity Con.
  8. ↑ http://catalog.asme.org/Codes/PrintBook/VV_10_2006_Guide_Verification.cfm

Evidence-based Uncertainty Quantification

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