# Probabilistic Methods in Structural Dynamics

#### Introduction

In structural mechanics and dynamics, engineers aim to approximate the behavior of a real world structure using mathematical models. These models are often built from partial differential equations, e.g. in Soil Dynamics or Acoustic Metamaterials. Often, the structural parameters entering these models, such as stiffnesses, masses or damping terms, are assumed to be deterministic. In reality however, this simplification might not be justified, since the parameters are not well known or identified. This uncertainty stems from a lack of knowledge of the parameter values and possibly to a lack of understanding of the behavior of the actual system. To account for this uncertainty, the problem can be stated in a probabilistic setting.

In general, we are interested in uncertainty quantification as well as as parameter or system identification via Bayesian updating for structural dyanmic models.

#### Uncertainty Quantification

Uncertainties in the model parameters or input will cause the model output to be uncertain as well. In the context of probabilistic uncertainty quantification the parameters are treated as random variables and probability distributions are assigned to them. In uncertainty quatnfiction one then tries to find the probability distribution or other relevant measures to describe the uncertainty in the model output. The knowledge about the uncertainty in a system's response is relevant for many applications, e.g. reliability assessment.

A number of methods have been developed to solve this problem. A straightforward approach is the Monte Carlo (MC) method. Other approaches, such as polynomial chaos expansion aim at approximating the model response as a function of the input parameters in a polynomial basis. For structural dynamic models described in the frequency domain, PCE has some limitiations. For that reason, we're interested in developing efficient surrogate models for such models. One approach is to make use of rational approximations composed of the ratio of the PCEs to describe the frequency response of a linear dynamic system, see [1]. In [1] a the coefficients in the PCEs are computed using a least-squares approach is presented. A further extension of the method is proposed in [2], where a sparse Bayesian learning approach for the computation of the surrogate model coefficients is proposed. The sparse Bayesian learning approach identifies the relevant basis vectors in both the numerator and denominator polynomials and can thus tacke the problem of overfitting in cases where only limited data from model evaluations is available.

#### Bayesian Updating

Additional to forecasting a system response using mathematical modeling, one can obtain measurement data from existing structures, whenever a structure has already been built. Quite often, the measured response and the model prediction are not consistent. That could be due to wrong parameters in the model or a model error. Using Bayesian updating, one can use the available measurement data to update the probabilistic description of the chosen input parameters and ideally reduce the uncertainty associated with them. This is also known as parameter updating. When dynamic response data is used, quite commonly modal data is used to describe the misfit between the measured and model response. Whenever the modal density is high, however, modal identification might be cumbersome or error prone. For these cases we are interested in frequency response data to describe said misfit. Our research focusses on formulating suitable error models that account for the correlation between observations to formulate the likelihood in the Bayesian approach, see [3].

#### Matlab-Toolbox for Rational Polynomial Chaos Expansion

We publish a toolbox for handling and computing rational PCE surrogate models on our LRZ-Git-repository.

#### Further Reading

[1] Schneider, Felix; Papaioannou, Iason; Ehre, Max; Straub, Daniel: Polynomial chaos based rational approximation in linear structural dynamics with parameter uncertainties. Computers & Structures **233**, 2020, 106223

[2] Schneider, Felix; Papaioannou, Iason; Müller, Gerhard: Sparse Bayesian Learning for Complex‐Valued Rational Approximations. International Journal for Numerical Methods in Engineering, 2022.

[3] Schneider, Felix; Papaioannou, Iason; Winter, Christoph; Straub, Daniel; Müller, Gerhard: Bayesian parameter updating in linear structural dynamics with frequency transformed data using rational surrogate models. Mechanical Systems and Signal Processing **166**, 2022