Gaussian Processes | igor kavrakov

Low-order Structural Aerodynamics

Fri, 17 Mar 2023 00:00:00 +0000

Fluid-structure interaction is a complex phenomenon that involves two distinct mediums: a fluid and a solid. Simulating fluid-structure interaction can be accomplished through solving the coupled partial differential equations of the fluid and solid by means of numerical methods. These techniques require discretizing the fluid equations on many degrees of freedom, be that vortex particles, volume cells, or finite elements, and usually require long computational times. When we are interested in the effect of the fluid on the structure, low-order aerodynamic models are like a secret weapon since they model the forces as a function of the motion of a 2D rigid deck (3 degrees of freedom) and free-stream turbulent fluctuations, thereby reducing the degrees of freedom of the fluid and increasing computational efficiency. This provides deeper insight into the underlying physics, helping us understand the factors contributing to the modeled phenomena. Aerodynamic forces are also one of the most important aspects when designing slender civil structures, such as bridges, towers and masts.

Low-order aerodynamic forces acting on a 2D ridgid deck

Aerodynamic force models can be broadly classified into two categories: semi-analytical and data-driven. Semi-analytical models are white-box models based on physics and use aerodynamic coefficients to substitude the fluid behavior. On the other hand, data-driven models are black-box models that use machine learning methods to reverse-engineer the relationship between input (motion/gusts) and output (forces) by using Computational Fluid Dynamics (CFD) or wind tunnel data. We can use both models to efficiently simulate structural response and capture various aerodynamic phenomena such as buffeting, flutter, and vortex-induced vibrations.

Aeroelastic phenomena: structural response against wind speed

I studied aerodynamic force models from multiple perspectives, including their intrinsic mathematical properties, qualitative comparison, and development of new models. Part of my research work is summarised here, namely:

On the assesment of aerodynamic models using Category Theory
On the influence of the assumptions of aerodynamic models on the aeroelastic response
On the data-driven aerodynamic modelling using Gaussian Processes and Neural Nets

Aerodynamic Modelling via Category Theory

Initially, we employed and extended an abstract framework based on Category Theory to evaluate the complexity of aerodynamic models and establish a basis for comparison among semi-analytical models based on their mathematical constructions (i.e., assumptions). Category theory, sometimes referred to as “abstract nonsense” by some, is a mathematical theory that deals with structures such as categories, objects, arrows, and the rules that relate them. Using this theory, we constructed a diagram defining aerodynamic model complexity. This immediately allows for comparisons between models without the need for prior knowledge of the semi-analytical models. For example, we cannot compare models if there is no direct diagramic relation between them (see below). More information on the modeling framework, including the definition of model comparability, simplest and most complex models, as well as practical considerations, can be found in our Royal Society Open Science paper.

Influence of Model Assumptions on the Aeroelastic Response

In addition to quantitative comparison, we examined the direct impact of aerodynamic assumptions on the buffeting response and critical flutter velocity. To do this, we compared the semi-analytical models to each other and to a CFD model by conducting a one-to-one comparison using identical wind turbulence input. The video below illustrates a one-to-one comparison between the Linear Unsteady (LU) semi-analytical model and the CFD model.

It turns out that, the fluid memory assumption, captured by the Linear Unsteady (LU) model, plays a crucial role in the aerodynamic response. This can be seen from the figures below, which depict the buffeting response. The LU and the hybrid nonlinear model (HNL) demonstrate the closest match to the CFD model, which is used as the reference due to its higher complexity (as previously depicted in the complexity diagram above).

2D buffeting analysis of the Great Belt Bridge deck: instantaneous velocity field (top); sample of response time-histories (centre); root-mean-square of the response for a selected wind speed range and turbulence intensity (bottom)

Additional comparisons can be found in our two papers, one in Engineering, and the other in Fluids and Structures. The first paper is based on a modified model of the Mersey Gateway Bridge (UK) (see video below) in its construction stage and it also presents a simplified method for employing the aerodynamic admittance in the time-domain based on Fourier Transform. The second paper is, to the best of my knowledge, the first study that conducts a comprehensive one-to-one comparison between CFD and semi-analytical models for buffeting and flutter analyses.

Dealing with time-histories (response or forces), comparisons based on the root-mean-square and peak values are not exhaustive to quantify the differences between models. Thus, we introduced a set of eight comparison metrics to quantify the discrepancies between two signals by looking at salient signal features relevant for structural aerodynamics. Each metric ranges from 0 to 1, with 1 indicating a perfect match. We propose eight metrics, including: phase, peak, root-mean-square, magnitude, wavelet and normalized wavelet (time-frequency), and bispectrum (second-order harmonic). For instance, the bispectrum metric quantifies the difference in second-order nonlinearities that may occur in the self-excited aerodynamic forces at large angles of attack. The comparison metrics are explained in detail in our Engineering Mechanics paper (incl. code - Github link).

Comparison metrics for time-histories of self-excited forces: small angles of attack (left); large angles of attack (right)

Data-driven Aerodynamic Models via Gaussian Processes and Neural Nets

Semi-analytical models are only as good as their mathematical properties, and it can be challenging to construct a white-box mathematical model that can capture memory-dependent nonlinearities and subsequently determine the corresponding aerodynamic coefficients (e.g., Volterra Series). To overcome this challenge, we employ data-driven techniques to reverse-engineer the model by learning the latent function that maps input (angle of attack) to output (aerodynamic force). We used both Gaussian Process (GP) Regression and Artificial Neural Networks (ANNs) as machine learning methods to construct the latent function; thereby, significantly increasing the mathematical capabilities of the model compared to semi-analytical models. The other key ingredient in constructing such model, which is fortunately abundant in wind engineering due to wind tunnel experiments, is data. Training the model requires appropriate data that reflects the use of the model. In practice, this means that the training signals should contain amplitudes and frequencies of the angle of attack that the bridge is expected to exhibit during flutter or buffeting. To address this, we developed a simple method based on Fourier transform that generates training signals for coupled motion.

Framework for data-driven modelling (top); Gaussian process model (bottom-left); artificial neural network model (bottom-right)

In our Journal of Wind Engineering and Industrial Aerodynamics paper (incl. code - Github link), we were able to train a GP model for self-excited forces, verify it using flat plate linear solutions, and predict second-order harmonics for Great Belt Bridge deck and Limit Cycle Oscillations (LCOs) for Tacoma-like deck. This is a promising development and we are continuing to explore this direction. However, we encountered an obstacle in predicting the LCO amplitude for coupled flutter, which we suspect is due to the non-stationary nature of the frequency content in the response. Specifically, at large amplitudes, the coupled frequency shifts towards torsional vibrations caused by large leading-edge separation. Similarly, we used standard ANNs in Computers and Structures paper and successfully predicted torsional flutter; however, we did not consider coupled excitation during the learning process in this study.

We are still exploring ways to use GPs as aerodynamic force models and incorporate physical principles into them. This research will eventually become a separate project - stay tuned!

Gaussian process aerodynamic force model: self-excited forces for large amplitude rotation of the Great Belt Bridge deck (top); limit cycle oscilation prediction of the rotation from free-vibration analysis of a Tacoma-like H-shaped deck (bottom)

System Identification

Mon, 16 Jan 2023 00:00:00 +0000

Noisy measurements from sensors can augment our mechanical models and improve our understanding of the actual state of our structures in reality: Are our input parameter and model assumptions correct or if there is any change in the system (possibly over time). This, in simple terms, represents the core objective of the Structural Health Monitoring (SHM) field, which aims to update and improve prediction models for structural assessment and maintenance. To achieve this goal, SHM employs different system identification and model updating techniques to learn from and incorporate noisy data into numerical and mechanical models. To this end, the use of machine learning methods as reverse-engineering probabilistic models has been quite beneficial in this field, particularly with recent advancements that have enabled the integration of these models with traditional mechanical models in a hybrid manner. This combination has led to a better understanding and improved prediction of the structural integrity of civil structures.

My research work in this field has primarily focused:

On the use of Gaussian Processes (GPs) in a physics-informed manner for parameter estimation and prediction of beams and plates
On the influence of material and geometric uncertainty on the dynamic properties of spun-cast prestressed concrete poles

On the phyisics-informed Gaussian Proceses for application to Kirchoff-Love plates and Timishenko Beams

This is an ongoing work that we primarily conduct with Gledson Tondo (a PhD student I co-advise). We employ Gaussian Processes (GPs) in a “physics-informed” setting (Raissi et al. 2017, J. Comp. Phys. 348) on the partial differential equations of Kirchhoff-Love plates and Timoshenko beams. The idea is to place a GP on the displacement field and apply the linear integro-differential operators of the mechanical plate/beam models to arrive at a probabilistic model that integrates both mechanical and data-driven models. Having noisy, heterogeneous data from sensors (e.g. strain and displacements), we are able to train this model while imposing physical relations between the measured quantities, and use it for prediction. In other words, we are able to infer probability distributions of the stiffness and physical quantities (e.g. displacements or internal forces) at places without any observations. The training is done using Markov Chain Monte Carlo sampling on the posterior of hyperparameters, yielding probabilistic estimates of both parameters and predictions. Additionally, we exploit the treatment of the physical quantities as probabilistic fields, and conditional entropy to develop an algorithm for sensor placement.

Two papers are in the works on this topic, one on Timoshenko beams (see this Paper), and another one is in works on Kirchhoff-Love plates. Below we show a couple of figures where the stiffness and prediction of physical quantities for both a plate and a beam are determined (the beam includes experiments!).

Physics-informed Gaussian Process of a Kirchhoff-Love plates: schematic of a fixed uniformly-loaded plate and learned stiffness (top); predicted displacement field (centre); predicted displacement at span (bottom-left) and at support (bottom-right)

Physics-informed Gaussian Process of a Timoshenko beam: experimental setup (top); various measurment sensors (centre-left); learned stiffness (centre-right); noisy measurments (bottom-left); predicted displacements (bottom-right)

On the influence of geometry and meterial parameters on the dynamic properties of concrete poles

Our Pole Team of Research Training Group 1462 conducted an investigation into the impact of geometric and material uncertainty on the numerical modeling of a spun-cast pre-stressed concrete pole. In our experimental study, we carried out various tests to compare the parameter uncertainty with design code values, including: i) laser-scanning of the geometry, ii) concrete compressive tests on drilled cores from the pole, and, iii) dynamic tests with hammer excitation. We found that the pole’s thickness was approximately 17% higher, while the concrete compressive strength and Young’s modulus were roughly 62% and 15% higher, respectively, than the design values. To study the influence of these discrepancies, we created a finite element model, and compared the eigenfrequencies and modes to those determined experimentally from the dynamic test. We found that using the experimental thickness and material in the numerical model, yielded improvement from 7.5% to 1.2% difference in the first eigenfrequency. More information can be found in our Bautechnik paper (in German).

Spun-cast pre-stressed pole experiments: dynamic tests (top); thickness measurments (centre-left); comparison designed vs. measured thickness (centre-right); numerical mode shape (bottom-left); comparisons first mode shape (bottom-right)