Authors: Owais Saleem, Afra Aksel

Supervisor: Felix Schneider

Introduction and Motivation

The goal of this project is to determine the modal quantities and the associated uncertainties in a 2-degree-of-freedom (2-DOF) frame system using surrogate modelling based on polynomial chaos expansion. In order to accomplish this, an analytical model of the system is developed, which is used to compute the modal quantities that will be utilized to train the surrogate model through polynomial chaos expansion (PCE). To incorporate randomness in the analytical model, the Monte Carlo Simulation technique is employed to generate random inputs, which follow a Gaussian (normal) distribution. In particular, the depth of the beams (d) and Young's Modulus (E) are chosen as the random input parameters for the analytical model of the 2-DOF system. The reason behind this choice is the knowledge that the depth parameter influences the mass of the system, whereas the Young's Modulus parameter affects the stiffness of the system

                                                                                                                         Representation of a 2-DOF frame system                                                                                   

Surrogate Modelling

Surrogate modelling, also known as meta-modelling, is a technique used in computational science and engineering to approximate complex or computationally expensive models. The idea behind surrogate modelling is to replace a computationally expensive model with a simpler, less expensive one that can be used to quickly make predictions and perform optimization and design exploration. The surrogate model is created by training a machine learning algorithm on a set of input-output pairs obtained from evaluating the complex model. The goal is to find a model that can predict the output of the complex model with a high degree of accuracy, but can be evaluated much faster.

Motivation for using surrogate model is that it provides quick, reliable estimates of the outputs of the complex model, without the need to run the full, computationally expensive simulation every time and provide a more intuitive understanding of the relationships between inputs and outputs in a complex model.

                               Scheme that describes the processes statistical surrogate modelling and analytical simulations  [F1]                                                                                                                                                                                                                                      

Polynomial Chaos Expansion (PCE)

Polynomial Chaos Expansion (PCE), is a type of surrogate modeling technique used in computational science and engineering. It is a mathematical method for representing the behavior of a complex model in terms of a polynomial expansion. The idea behind PCE is to represent the output of a complex model as a sum of polynomial terms. The coefficients of the polynomial are determined through statistical regression, using data obtained by evaluating the complex model at various input points. The resulting polynomial is then used as the surrogate model, which can be evaluated much faster than the original model. PCE is particularly well-suited for problems with uncertain inputs, such as those involving random variables. The polynomial terms in the PCE representation can capture the effects of uncertainty, allowing the surrogate model to be used for sensitivity analysis, uncertainty quantification, and optimization under uncertainty

Orthogonality of Polynomials  means:

$//<![CDATA[ \begin{array}{l}\[ \int_{}^{} P_i(x) P_j(x) w(x) \,dx \] = h_i \delta_i_j\end{array} //]]>$ where w(x)  the weigh functions and $//<![CDATA[ \begin{array}{l}\delta\end{array} //]]>$ is the kronecker delta.

                                                                                                                      Orthogonal polynomial classes for random variable types   [F2]

Frequency Response Function

Frequency Response Function (FRF) is a measure of the system's response to sinusoidal excitation (input) at different frequencies. The FRF is a transfer function that describes the relationship between the input and output of the system in frequency domain and is used to quantify the dynamic behavior of the system. FRFs can be measured experimentally by applying a sinusoidal input to the system and measuring the resulting output at different frequencies. Alternatively, they can be calculated mathematically based on a system's physical parameters and properties. The formula used to calculate FRF function H($//<![CDATA[ \begin{array}{l}\Omega\end{array} //]]>$) from the analytical model is: 

$//<![CDATA[ \begin{array}{l}H(\Omega)= (K-\omega^2M+i\omega C)^{-1}\end{array} //]]>$ where M is the mass matrix, K is the stiffness matrix, C is the (Reyleigh) damping matrix and $//<![CDATA[ \begin{array}{l}\Omega\end{array} //]]>$ is the excitation frequency  

                                                                                                                                     Demonstration of frequency response [F3]

Uncertainty Quantification and Propagation

Uncertainty quantification (UQ) is a field of computational sciences that focuses on the study of the effects of uncertainties in input parameters on the outputs of complex models. In UQ, input parameters are typically represented by random variables based on available data and expert knowledge, and the goal is to propagate these uncertainties through the model to obtain estimates of the statistics of the quantities of interest. Monte Carlo simulation (MCS) is a popular technique for UQ, but it can be computationally expensive when each model run is time-consuming. To address this issue, alternative techniques such as surrogate models have been developed.

Uncertainties in computer simulations can come from various sources such as incomplete information about physical effects, incomplete model parameters, and limited knowledge of the system being simulated. These uncertainties can be divided into two categories: aleatoric and epistemic. Aleatoric uncertainty is caused by random variations and can be represented by probability distributions. Epistemic uncertainty is caused by incomplete knowledge of the system and can be reduced by gathering more information about it. This type of uncertainty may come from human errors, experimental design, or equipment calibration.

Deterministic models ignore uncertainties in parameters and their effects on responses. To make robust decisions, it's necessary to quantify these uncertainties and propagate them through the model. Uncertainty quantification (UQ) is focused on quantifying the amount of uncertainty present in a system or result, while uncertainty propagation (UP) is focused on understanding how that uncertainty propagates through a system.

                                                                                                                                                                                          Demonstration of UQ and UP  [F4]

Random variables (E & d)

The Nataf transformation, also known as the Gaussian copula, is a statistical method used to model the dependence between multiple random variables which is highly relevant in generating random variables in the course of this project. The method requires the definition of the marginal distribution which describes the distribution of a single random variable and correlation matrix which describes the relationships between multiple variables. ERADist is a tool developed at TUM for creating and operating distribution functions.

As mentioned earlier random variables used in the project and their properites are as:

The distribution can be generated with the above mentioned tools as:

The plotted distributions look like:

                                                                                                                                                     Lognormal distribution beam depth (left) and Youngs Modulus (right)

Sample Generation

• Using Latin Hypercube Sampling (LHS) technique to generate random uniformly distributed samples from 0 to 1. This is important in order to have samples from the entire domain of the random variables.
• Then getting the samples from uniform space (0-1) to normal space
• After that getting samples from real space using Nataf transformation.

For every random variable of a Monte Carlo simulation model, samples are taken from Uniform(0,1) distributions, so each generated scenario is just as likely to occur as any other. 

However, due to the shape of each cumulative curve, more values will be generated where the cumulative curve is at its steepest.

                                                                                              LHS and norminv demonstrated (left)                                                                                 Sample generation explained on cdf (right)[F5]

,

MCS vs PCE

Monte Carlo Simulation (MCS) is a conventional but an effective and accurate way of propagating uncertainty through a model. However with increasing complexity, it may not be feasible. But once our PCE model is ready, it can quantify uncertainty in one run.

Below is the complete illustration:

                                                                                                                                            Calling the function/model for each sample-MCS [F6]                                        Uncertainty propagation in one step- PCE [F7]

PCE PARAMETERS,TRAINING & TESTING

The number of unknown coefficients in a Polynomial Chaos Expansion (PCE) depends on the number of basis functions used to represent the expansion and the dimensionality of the input space. Assuming a PCE with N basis functions and d input dimensions, the total number of unknown coefficients is given by the expression:

$//<![CDATA[ \begin{array}{l}\frac{ (N+ d-1)!}{N!* (d-1)!}\end{array} //]]>$

Schematic of PCE Modelling

The process of training and validating a surrogate model using Polynomial Chaos Expansion (PCE) begins with the generation of a list of training samples. The analytical model is then invoked to obtain the true or reference Eigen values and Eigen vectors, which are utilized in the training process.

The PCE model is trained using the training samples and the reference Eigen values and Eigen vectors. Subsequently, a new batch of samples is generated and the PCE model is utilized to obtain the output values for these samples. The analytical model is not invoked during this stage of the process.

To evaluate the performance of the PCE model, a response comparison curve is plotted to compare the output values obtained from the PCE model with the expected values obtained from the analytical model. This enables an assessment of the accuracy of the surrogate model and the identification of areas where improvements may be necessary.

NOTE: Since the frame system in this project is a 2 DOF system, two PCE models for 2 eigen values and 4 PCE models for 4 entries of mass normalized eigen vectors are created.

Result Analysis

Model Comparison

After successfully training the PCE model (polynomial degree =5)  with 30 training samples, the test results were as

                                                                                           The four plots show that the model performs well (all outcomes lie on the identity line) in predicting response for modal quantities with the given parameters.

FRF through PCE vs FRF through Analytical Model

Once we have eigen frequencies and eigen vectors generated through PCE model from real samples of Youngs Modulus (E) and depth of floor/beam(d). We use those values to generate FRF matrix using the formula:

$//<![CDATA[ \begin{array}{l}H(\Omega) = \sum_{i=1}^{n} \frac{ \phi_i \phi_i^{T}} {\omega_i^2 -\Omega^2+2i\Omega \omega_i \xi_i}\end{array} //]]>$ where $//<![CDATA[ \begin{array}{l}\Omega\end{array} //]]>$ is the excitation frequency, $//<![CDATA[ \begin{array}{l}\phi_i\end{array} //]]>$ is the eigen vector corresponding to natural frequency $//<![CDATA[ \begin{array}{l}\omega_i\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}\xi_i\end{array} //]]>$ is its modal damping ratio.

For one realization of <E,d> we predict PCE modal quantities  $//<![CDATA[ \begin{array}{l}\omega_i\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\phi_i\end{array} //]]>$. Then varying the excitation frequency $//<![CDATA[ \begin{array}{l}\Omega\end{array} //]]>$, we generate one plot of FRF matrix.

And for the same input values of <E,d> we generate one plot of FRF matrix using our Analytical model. And comparing the two plots together depicts the accuracy of PCE because FRF from Analytical model through MCS serves as the benchmark. Below are some of the plots that we have generated:

X1 corresponds to response at DOF 1 and X2 corresponds to response at DOF 2

If  let's say FRF(1,1) which means response at DOF 1 when excited at DOF 1 are generated  using PCE and analytical model on the same plot, then they should coincide for the same input <E,d> and excitation fequency $//<![CDATA[ \begin{array}{l}\Omega\end{array} //]]>$.

That's what has been received as shown below:

Capturing uncertainty with PCE:

The most exciting part was propagating uncertainty from <E,d> to modal quantities and then to FRF and capturing that in one plot. 

10 value pairs of <E,d> are chosen and hence 10 pairs values of eigen values $//<![CDATA[ \begin{array}{l}\omega_i\end{array} //]]>$, 10 eigen matrices $//<![CDATA[ \begin{array}{l}\Phi\end{array} //]]>$ . Going through each one of them in a loop ten times on the same plot gives the result as:

Uncertainty in the eigen values associated with uncertainty in input parameters <E,d>

Parametric Influence on performance of PCE

PCE involves approximating the output of a system as a polynomial function of its input parameters. The coefficients of the polynomial are determined through a series of operations, including orthogonal projection of the input parameters onto a set of basis functions.

The influence of the input parameters on the PCE approximation can vary based on several factors, including the order of the PCE, the choice of basis functions, and the nature of the system being studied. Generally, the higher the order of the PCE, the more accurately it can capture the behavior of the system. However, higher-order PCEs can also require more computational resources to calculate, and can be more sensitive to changes in the input parameters.

The choice of basis functions can also affect the accuracy of the PCE approximation. Different basis functions are suited to different types of problems, and the choice of basis functions can influence the accuracy and computational efficiency of the PCE. For example, Hermite polynomials are often used for systems with normally distributed input parameters, while Legendre polynomials are more suitable for uniformly distributed input parameters.

The nature of the system being studied can also influence the influence of input parameters on the PCE approximation. For example, if some input parameters have a stronger influence on the output than others, the PCE approximation may be more sensitive to changes in those parameters. Additionally, if the system exhibits nonlinearity or other complex behavior, the accuracy of the PCE may be affected, as the polynomial approximation may not be able to capture the full range of system behavior.

In summary, the influence of input parameters on the PCE approximation depends on several factors, including the order of the PCE, the choice of basis functions, and the nature of the system being studied. By carefully selecting the appropriate PCE parameters for a given problem, researchers can obtain accurate and efficient uncertainty quantification results.

Performance can also be improved by increasing model flexibility in case of underfitting or increasing sample space in case of overfitting.

                                                                                                                            Demonstration of underfitting, overfitting and balanced cases for parametric influence [F8]

Underfitting:

Here is an example of underfitting and computing ideal polynomial degree by minimizing error in eigen values:

So when the degree of polynomials is 3, the deviation tends to zero.

                                                                                                                        Typical case of the underfitting                                      Increasing the polynomial degree to optimum to avoid underfitting

Overfitting:

If the polynomial degree is exceedingly high or the number of training data points is low, the model tries to interpolate all the training points which gives a very irregular behavior on test  samples because of high oscillations.

No. of unknown coefficients in both cases is 21 but no. of training samples is just 20 for the second case(right) which results in overfitting.

Wrong data sampling:

The data that we use to train our PCE model should be representative of the entire space of random variables. If not, the errors are bound to occur, though we may not be able to always locate the errors.

                                                                   The deviation at the starting of the axis shows that the model has not seen samples like this and hence the prediction is poor.

Conclusions:

The fact that the analytical model and the surrogate model results coincide indicates that the surrogate model is a reliable and accurate representation of the system's behavior. This finding suggests that the surrogate model can be used as an effective tool for predicting the response of the frame system to a range of excitations in a computationally efficient manner. Overall, these results contribute to a better understanding of the dynamic behavior of the frame system and can inform the design of more efficient and reliable structures. The use of surrogate modelling based on polynomial chaos expansion in this project provides an efficient and accurate method for determining the modal properties and uncertainties associated with the 2-DOF frame system. The insights obtained from this study can be utilized to enhance the design of such systems, optimize their performance, and improve their reliability and durability. The uncertainty propagation of modal variables obtained through PCE modelling is fairly accurate as well.

It is important to mention here that for a small project like 2 DOF system, it doesn’t make much sense to use PCE because the computational time using original model is more or less the same but for a large project it can be very convenient because inverse operations are very expensive or likewise calling a FEM for modal quantities. The model accuracy highly depends on the parameters for building a PCE model. Therefore convergence studies must be performed for the optimal polynomial degree to minimize errors and avoid overfitting or underfitting of the model.

References:

Sources:

[1] Schneider, F., Papaioannou, I., & Müller, G. (n.d.). Sparse Bayesian Learning for Complex-Valued Rational Approximations. aEngineering Risk Analysis Group - Technical University of Munich, Arcisstr. 21, 80333 Munich, Germany bChair of Structural Mechanics, Technical University of Munich, Arcisstr. 21, 80333 Munich, Germany.

[2] Sullivan, T. J. (n.d.). Introduction to Uncertainty Quanti cation (Vol. 63). Springer. https://www.springer.com/series/1214

[3]  Risk Analysis Notes – Engineering Risk Analysis Group, TUM

[4] Surrogate model. (n.d.). Wikipedia. Retrieved February 19, 2023, from https://en.wikipedia.org/wiki/Surrogate_model

[5] Chesser, L. (2020, October 29). An introduction to Surrogate modeling, Part I: fundamentals | by Shuai Guo. Towards Data Science. Retrieved February 19, 2023, from https://towardsdatascience.com/an-introduction-to-surrogate-modeling-part-i-fundamentals-84697ce4d241

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[7] [1810.09116] Surrogate modeling based on resampled polynomial chaos expansions. (2018, October 22). arXiv. Retrieved February 19, 2023, from https://arxiv.org/abs/1810.09116

[8] Polynomial Chaos Expansion — Uncertainty Quantification. (n.d.). Uncertainty Quantification. Retrieved February 19, 2023, from https://dictionary.helmholtz-uq.de/content/polynomialChaosExpansion.html

[9] Papaioannou, I. (2318). Stochastic Finite Element Methods. Engineering Risk Analysis Group, TUM.

[10] An Efficient Polynomial Chaos Expansion Method for Uncertainty Quantification in Dynamic Systems. (n.d.). MDPI. Retrieved February 19, 2023, from https://www.mdpi.com/2673-3161/2/3/26

[11] B. Sudret, S. Marelli and J. Wiart, "Surrogate models for uncertainty quantification: An overview," 2017 11th European Conference on Antennas and Propagation (EUCAP), Paris, France, 2017, pp. 793-797, doi: 10.23919/EuCAP.2017.7928679.

Figures :

[F1]  Chesser, L. (2020, October 29). An introduction to Surrogate modeling, Part I: fundamentals | by Shuai Guo. Towards Data Science. Retrieved February 19, 2023, from https://towardsdatascience.com/an-introduction-to-surrogate-modeling-part-i-fundamentals-84697ce4d241 

[F2] Kumthekar, A. (n.d.). Uncertainty Quantification of the lifetime of self-healing thermal barrier coatings based on surrogate modelling of thermal cyclic fracture and healing. Thesis-TU Delft.

[F3] What is a Frequency Response Function (FRF)? (2020, July 10). Siemens Communities. Retrieved February 19, 2023, from https://community.sw.siemens.com/s/article/what-is-a-frequency-response-function-frf

[F4] B. Sudret, S. Marelli and J. Wiart, "Surrogate models for uncertainty quantification: An overview," 2017 11th European Conference on Antennas and Propagation (EUCAP), Paris, France, 2017, pp. 793-797, doi: 10.23919/EuCAP.2017.7928679.

[F5] What is Monte Carlo simulation? (n.d.). Vose Software. Retrieved February 19, 2023, from https://www.vosesoftware.com/riskwiki/MonteCarloSimulationintroduction.php 

[F6] Risk Analysis Notes – Engineering Risk Analysis Group, TUM

[F7] Schneider, F., Papaioannou, I., & Müller, G. (2022, February 7). Sparse Bayesian Polynomial Chaos Expansion for Uncertainty Quantification of Frequency Response Functions Computational Methods in Systems and Control Theory Seminar. Chair of Structural Mechanics Technical University of Munich - Engineering Risk Analysis Group Technical University of Munich.

[F8] Model Fit: Underfitting vs. Overfitting - Amazon Machine Learning. (n.d.). AWS Documentation. Retrieved February 19, 2023, from https://docs.aws.amazon.com/machine-learning/latest/dg/model-fit-underfitting-vs-overfitting.html