科学计算，仿真，CAE的VVUQ
n5321 | 2025年7月28日 17:01
Define： We will refer to the application of a model to produce a result, often including associated numerical approximation errors, as a simulation。mathematical models that take the form of coupled systems of nonlinear partial differential equations. （用非线性偏微分方程描述的数学模型）
目的：understand，predict behavior and optimize Performance！predicting the behavior of natural and engineered systems
问题：是中间存在不确定，所以结果不准确，所以需要VVUQ. a fundamental disconnect often exists between simulations and practical applications.
解决方式：Information on the magnitude, composition, and sources of uncertainty in simulations is critical in the decision-making process for natural and engineered systems.
控制参数：system response quantities
负面后果：decision makers will be ill advised, possibly resulting in inadequate safety, reliability, or performance of the system. Consequently, decision makers could unknowingly put at risk their customers, the public, or the environment. 哥伦比亚号，福岛核电站。
正面后果：CBA（certificate by analysis）
源头：uncertainty (aleatory and epistemic) ：In scientific computing, there are many sources of uncertainty including the model inputs, the form of the model, and poorly characterized numerical approximation errors. All of these sources of uncertainty can be classified as either purely aleatory, purely epistemic, or a mixture of the two.
manufacturing processes, natural material variability, initial conditions, wear or damaged condition of the system, and the system surroundings. the modeling process itself can introduce large uncertainties due to the assumptions in the model（model validation ） as well as the numerical approximations （code and solution verification ）employed in the simulations.
错误类别分类一
Aleatory (random) uncertainties in model inputs are treated as random variables, the stochastic process can be characterized via a probability density distribution
Aleatory uncertainty is generally characterized by either a probability density function (PDF) or a cumulative distribution function (CDF).
epistemic uncertainty (also called reducible uncertainty or ignorance uncertainty) due to lack of knowledge by the modelers, analysts conducting the analysis, or experimentalists involved in validation. a lack of knowledge on the part of the analyst, or team of analysts, conducting the modeling and simulation.
The lack of knowledge can pertain to, for example, modeling of the system of interest or its surroundings, simulation aspects such as numerical solution error and computer round-off error, and lack of experimental data.
If knowledge is added (through experiments, improved numerical approximations, expert opinion, higher fidelity physics modeling, etc.) then the uncertainty can be reduced.
错误源头（model inputs, numerical approximations, and model form uncertainty ）
Model inputs
Model inputs include not only parameters used in the model of the system, but also data from the description of the surroundings (see Figure 1). Model input data includes things such as geometry, constitutive model parameters, and initial conditions, and can come from a range of sources including experimental measurement, theory, other supporting simulations, or expert opinion. Data from the surroundings include boundary conditions and system excitation (e.g., mechanical forces or moments acting on the system, force fields such as gravity and electromagnetism).
Numerical approximation
Since differential equation-based models rarely admit exact solutions for practical problems, approximate numerical solutions must be used. The characterization of the numerical approximation errors associated with a simulation is called verification. It includes discretization error, iterative convergence error, round-off error, and also errors due to computer programming (coding mistakes) mistakes.
Figure 2 depicts the propagation of input uncertainties through the model to obtain output uncertainties. The number of individual calculations needed to accurately accomplish the mapping depends on four key factors: (a) the nonlinearity of the partial differential equations, (b) the dependency structure between the uncertain input quantities, (c) the nature of the uncertainties, i.e., whether they are aleatory, epistemic, or mixed uncertainties, and (d) the numerical methods used to compute the mapping.
Model form
The form of the model results from all assumptions, conceptualizations, abstractions, approximations, and mathematical formulations on which the model relies.
The characterization of model form uncertainty is commonly estimated using model validation.
assessment of model accuracy by way of comparison of simulation results with experimental measurements.
(a) assumptions concerning the environment (normal, abnormal, or hostile) to which the system is exposed,
(b) assumptions concerning the particular scenarios the system is operating under, e.g., various types of damage or misuse of the system, and
(c) cases where experimental data are not available on any closely related systems, e.g., data are only available on subsystems,
结果：
存在Errors
Numerical approximation errors （verification techniques ）
due to discretization, iteration, and computer round off
Model form uncertainty is quantified using
(a) model validation procedures statistical comparisons of model predictions to available experimental data
(b) extrapolation of this uncertainty structure to points in the application domain
procedure：
(1) the identification of all sources of uncertainty,
The goals of the analysis should be the primary determinant for what is considered as fixed versus what is considered as uncertain.
(2) characterization of model input uncertainties,
(a) assigning a mathematical structure to describe the uncertainty and
(b) determining the numerical values of all of the needed parameters of the structure.
Stated differently, characterizing the uncertainty requires that a mathematical structure be given to the uncertainty and all parameters of the structure be numerically specified such that the structure represents the state of knowledge of every uncertainty considered.
(3) elimination or estimation of code and solution verification errors, （Estimate uncertainty due to numerical approximations ）
Recall that verification deals with estimating numerical errors which include discretization error, iterative error, round-off error, and coding mistakes.
(4) propagation of input uncertainties through the model to obtain uncertainties in the SRQs,
to determine the effects on the SRQs.
The simplest approach for propagating aleatory uncertainty through a model is sampling.
(5) quantification of model form uncertainty
Model form uncertainty is estimated through the process of model validation.
First, we quantitatively estimate the model form uncertainty at the conditions where experimental data are available using a mathematical operator referred to as a validation metric.
Second, we extrapolate the uncertainty structure expressed by the validation metric to the application conditions of interest.
A validation metric is a mathematical operator that requires two inputs: the experimental measurements of the SRQ of interest and the prediction of the SRQ at the conditions used in the experimental measurements.
Model extrapolation: Numerous validation experiments would normally be required in order to estimate the area validation metric over the entire space of model input parameters for the application of interest. (In many cases, however, it is not possible to obtain experimental data at the application conditions of interest. ) The general process for determining the model form uncertainty at the conditions of interest (i.e., the prediction location) is as follows.
First, a regression fit of the validation metric is performed in the space of the validation domain.
Next, a statistical analysis is performed to compute the prediction interval at the conditions of interest.
In the past, it has been common practice to either (a) ignore the model form uncertainty in the predictions for the application conditions or (b) calibrate adjustable parameters in the mathematical model so that improved agreement could be obtained with the available experimental data at conditions “V”.
