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Uncertainty Quantification Module

Understand and Characterize Model Uncertainty

The Uncertainty Quantification Module is used for understanding the impact of model uncertainty — how the quantities of interest depend on variations in the inputs of a model. It provides a general interface for screening, sensitivity analysis, uncertainty propagation, reliability analysis, and inverse uncertainty quantification.

The Uncertainty Quantification Module can efficiently test the validity of model assumptions, convincingly simplify models, understand the key input to the quantities of interest, explore the probability distribution of the quantities of interest, discover the reliability of a design, and calibrate the unknown input parameter distributions. The assurance of model correctness and increased understandings of the quantities of interest aid in reducing costs in production, development, and manufacturing.

The Uncertainty Quantification Module can be used with products throughout the COMSOL product suite for analyzing uncertainties in electromagnetics, structural, acoustics, fluid flow, heat, and chemical engineering simulations. It can be combined with the CAD Import Module, Design Module, or any of the LiveLink™ products for CAD.

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The COMSOL Multiphysics UI showing some Uncertainty Quantification study results, with a Sobol index plot, Kernel Density Estimation graph, as well as a Confidence interval table.
A close-up view of the settings for input parameters and quantities of interest for a COMSOL Multiphysics model.

Input Parameters and Quantities of Interest

When running an uncertainty quantification study, a set of global quantities of interest is defined in terms of a COMSOL Multiphysics® model solution. As a result, the quantities of interest are functions of the input parameters. A set of study-dependent quantities of interest — for instance, time- or frequency-dependent expressions — can also be defined, in which case, the quantities of interest are functions of both the input parameters and the study-dependent parameters.

In the case of a structural analysis, the quantities of interest may be the maximum displacement, stress, or deflection angle. For a heat transfer or CFD analysis, the quantities of interest may be maximum temperature, total heat loss, or the total fluid flow rate. For an electromagnetics simulation, they may be resistance, capacitance, or inductance. For a MEMS simulation, they may include the frequency-dependent output power from a piezoelectric device. Since the Uncertainty Quantification Module is applicable to any physics model computed with the COMSOL Multiphysics® software, as well as any mathematical expression of various solved-for field quantities, the choices for the quantity of interest are endless.

Any uncertain model input, whether it be a physics setting, geometric dimension, material property, or discretization setting, can be treated as an input parameter, and any model output can be used to define the quantities of interest. The input parameters can be sampled analytically with probability distributions or with user-specified data. The analytically sampled input parameters can be correlated and uncorrelated, where the correlated input parameters can be grouped into correlation groups and sampled with the Gaussian copula method.

Screening

The Screening, MOAT study type implements a lightweight global screening method that gives a qualitative measure of the importance of each input parameter. The method is purely sample based, using the Morris one-at-a-time (MOAT) method, and requires a relatively small number of COMSOL model evaluations. This makes it an ideal method when the number of input parameters is too large to allow more computationally expensive uncertainty quantification studies.

For each quantity of interest, this MOAT method computes the MOAT mean and MOAT standard deviation for each input parameter. These values are presented in a MOAT scatter diagram. The ranking of the MOAT mean and MOAT standard deviations gives the relative importance of the input parameters. A high value of the MOAT mean implies that the parameter is significantly influencing the quantity of interest. A high value of the MOAT standard deviation implies that the parameter is influential and that it is either strongly interacting with other parameters or that it has a nonlinear influence, or both.

The COMSOL Multiphysics UI showing the Model Builder with the Uncertainty Quantification node highlighted, the corresponding settings, and a MOAT scatter diagram in the Graphics window.
A Sobol index plot for a quantity of interest, where x1 is shown as blue bars, x2 as green, x3 as red, and x4 as teal.

Sensitivity Analysis

The Sensitivity Analysis study type is used to compute how sensitive the quantities of interest are with respect to the input parameters. This study type includes two methods: the Sobol and correlation methods.

The Sobol method analyzes the entire input-parameter distribution and decomposes the variance of each quantity of interest into a sum of contributions from the input parameters and their interactions.

For each input parameter, the Sobol method computes the Sobol indices. The first-order Sobol index shows the variance of a quantity of interest attributed to the variance of each input parameters individually. The total Sobol index shows the variance of a quantity of interest attributed to the variance of each input parameters and its interaction with the other input parameters. The Sobol indices for each quantity of interest and all parameters are presented in a dedicated Sobol plot where the histograms are ordered by total Sobol index. The quantity of interest is most sensitive to the input parameter with the highest total Sobol index. The difference between the total Sobol index and first-order Sobol index for an input parameter measures the effect of the interaction between this input and others.

Compared to the screening method, sensitivity analysis is used to quantitatively analyze how uncertainties in the quantities of interest apportion to the different input parameters. This method requires more computational resources since the computation of accurate Sobol indices relies on a high-quality surrogate model.

The correlation method computes the linear and monotonic relationship between each input parameters and the quantities of interest. For sensitivity analysis based on the correlation method, four types of correlations are computed: bivariate, ranked bivariate, partial, or ranked partial correlation.

Uncertainty Propagation

The Uncertainty Propagation study type is used to analyze how the uncertainties of input parameters propagate to each quantity of interest by estimating their probability density function (PDF). The underlying physics that maps the input parameters to the quantities of interest through COMSOL Multiphysics® model evaluations is for most applications impossible to compute analytically.

For this reason, a Monte Carlo analysis is necessary to approximate the PDFs. Similar to the Sobol method, a surrogate model is used to dramatically reduce the computational cost of the Monte Carlo analysis. For each quantity of interest, a kernel density estimation (KDE) is performed and visualized as a graph, as an approximation of the PDF. Furthermore, based on this analysis, a confidence interval table provides, for each quantity of interest, the mean, standard deviation, minimum, maximum, and lower and upper bound values corresponding to confidence levels of 90%, 95%, and 99%. For each study-dependent quantity of interest, a plot of the mean with a 95% prediction interval is presented to show how the uncertainty of the quantity of interest varies with the study-dependent parameters.

A kernel density estimation line graph with KDE as the y-axis and Predicted as the x-axis.
A 1D plot showing two quantities of interest and a shaded region where the input parameters satisfy the reliability criteria.

Reliability Analysis

Compared to other uncertainty quantification study types, which investigate the overall uncertainty of the quantities of interest, the Reliability analysis, EGRA method addresses a more direct question. Given a nominal design and some specific uncertain inputs, what is the probability that the design fails? The failure can be a complete breakdown of the design, but it can also be phrased in terms of a quality criterion.

To ensure the reliability, the traditional approach of modeling and simulation is to use safety margins and worst-case scenarios. With a proper reliability analysis, it is possible to avoid overestimation and underestimation, since estimates of the actual probability can be made. A rough estimate can be drawn from the confidence interval table from the uncertainty propagation for each quantity of interest. But with reliability analysis, a more sophisticated reliability criteria can be defined based on combinations of the quantities of interest and corresponding thresholds. The efficient global reliability analysis (EGRA) method used for the reliability analysis study efficiently directs the computational resources to the limit state that separates the failure and success of the design.

Surrogate Models and Response Surfaces

Sensitivity analyses computed with the Sobol method, uncertainty propagation, and reliability analysis all rely on an accurate Monte Carlo-type analysis. This often requires a large number of evaluations to achieve good accuracy. For realistic problems where a COMSOL Multiphysics® model evaluation might require significant resources and where the uncertainty quantification analysis involves several parameters, a Monte Carlo analysis conducted only with COMSOL Multiphysics® model evaluations is computationally infeasible. A key feature of the Uncertainty Quantification Module is its ability to train and use a so-called surrogate model, also known as a metamodel, for a particular UQ analysis to save computational resources.

A surrogate model is a compact mathematical model constructed to represent and evaluate the quantities of interest in the domain of interest defined by the input parameters. This model is completely independent of the underlying COMSOL Multiphysics® model and can, when trained properly, be used instead of the COMSOL Multiphysics® model to predict values for the quantities of interest for other values of the input parameters than those solved for. The process of constructing a surrogate model is typically adaptive and the surrogate model can approximate the original model to a high degree of accuracy. With user-defined tolerances, the accuracy of the surrogate models can be increased. A higher level of accuracy requires additional COMSOL Multiphysics® model evaluations.

Once a surrogate model has been built, an independent verification can be performed to further test its validity, and response surface data can easily be computed for the entire input parameters space. A response surface can then be visualized, where one quantity of interest is plotted versus two input parameters at a time. The surrogate models are available as global functions for general use.

A response surface plot in the Rainbow color table.
Six plots displayed in a stacking sequence, with three plots in the Rainbow color table.

Inverse Uncertainty Quantification

Inverse uncertainty quantification (inverse UQ) is used when some of the input parameters have unknown probability distributions, known as calibration parameters. Using inverse UQ, experimental data can be propagated backward to gain insight into the statistical properties of these calibration parameters. In order to apply inverse UQ, a prior probability distribution is required for each calibration parameter before the analysis can be conducted.

Experimental data is typically available for the quantities of interest and the parameters used in experiments. There are also calibration input parameters that cannot be directly measured. For instance, consider an experiment where we want to calibrate the Young's modulus of a mechanical part. We should conduct an experiment that measures the tensile stress as a function of the specified material displacement. An inverse UQ study should then be set up to use the experimental data and prior knowledge of the Young's modulus to calibrate the probability distribution that would best reproduce the measured values of the tensile stress from the experiments. Inverse UQ can be applied to a wide range of physics-based models, including those related to structural mechanics, fluid flow, acoustics, heat transfer, electromagnetics, and chemical engineering.

To make the computation of the calibration parameters' posterior probability distributions feasible, a surrogate model is used together with a Markov chain Monte Carlo (MCMC) method. After the computation, the joint and marginal probability distributions of the calibrated input parameters can be visualized. Additionally, a confidence interval table is generated that provides information such as the mean; the standard deviation; the minimum and maximum values; and the lower- and upper-bound values corresponding to confidence levels of 90%, 95%, and 99% for each calibrated input parameter.

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