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Small- and large-scale characterization and mixing properties in a thermally driven thin liquid film
(2015)
We study aqueous, freestanding, thin films stabilized by a surfactant with respect to mixing and dynamical systems properties. With this special setup, a two-dimensional fluid can be realized experimentally. The physics of the system involves a complex interplay of thermal convection and interface and gravitational forces. Methodologically, we characterize the system using two classical dynamical systems properties: Lyapunov exponents and entropies. Our experimental setup produces convection with two stable eddies by applying a temperature gradient in one spot that yields weakly turbulent mixing. From dynamical systems theory, one expects a relation of entropies, Lyapunov exponents, a prediction with little experimental support. We can confirm the corresponding statements experimentally, on different scales using different methods. On the small scale the motion and deformation of fluid filaments of equal size (color imaging velocimetry) are used to compute Lyapunov exponents. On the large scale, entropy is computed by tracking the left-right motion of the center fluid jet at the separatrix between the two convection rolls. We thus combine here dynamical systems methods with a concrete application of mixing in a nanoscale freestanding thin film.
We present a novel experimental setup to investigate two-dimensional thermal convection in a freestanding thin liquid film. Such films can be produced in a controlled way on the scale of 5-1000 nm. Our primary goal is to investigate convection patterns and the statistics of reversals in Rayleigh-Benard convection with varying aspect ratio. Additionally, questions regarding the physics of liquid films under controlled conditions can be investigated, like surface forces, or stability under varying thermodynamical parameters. The film is suspended in a frame which can be adjusted in height and width to span an aspect ratio range of Gamma = 0.16-10. The top and bottom frame elements can be set to specific temperature within T = 15 degrees C to 55 degrees C. A thickness to area ratio of approximately 108 enables only two-dimensional fluid motion in the time scales relevant for turbulent motion. The chemical composition of the film is well-defined and optimized for film stability and reproducibility and in combination with carefully controlled ambient parameters allows the comparison to existing experimental and numerical data. Published by AIP Publishing.
The maximum entropy method is used to predict flows on water distribution networks. This analysis extends the water distribution network formulation of Waldrip et al. (2016) Journal of Hydraulic Engineering (ASCE), by the use of a continuous relative entropy defined on a reduced parameter set. This reduction in the parameters that the entropy is defined over ensures consistency between different representations of the same network. The performance of the proposed reduced parameter method is demonstrated with a one-loop network case study.
The maximum entropy method is used to derive an alternative gravity model for a transport network. The proposed method builds on previous methods which assign the discrete value of a maximum entropy distribution to equal the traffic flow rate. The proposed method however, uses a distribution to represent each flow rate. The proposed method is shown to be able to handle uncertainty in a more elegant way and give similar results to traditional methods. It is able to incorporate more of the observed data through the entropy function, prior distribution and integration limits potentially allowing better inferences to be made.
A maximum entropy (MaxEnt) method is developed to predict flow rates or pressure gradients in hydraulic pipe networks without sufficient information to give a closed-form (deterministic) solution. This methodology substantially extends existing deterministic flow network analysis methods. It builds on the MaxEnt framework previously developed by the authors. This study uses a continuous relative entropy defined on a reduced parameter set, here based on the external flow rates. This formulation ensures consistency between different representations of the same network. The relative entropy is maximized subject to observable constraints on the mean values of a subset of flow rates or potential differences, the frictional properties of each pipe, and physical constraints arising from Kirchhoff’s first and second laws. The new method is demonstrated by application to a simple one-loop network and a 1,123-node, 1,140-pipe water distribution network in the suburb of Torrens, Australian Capital Territory, Australia.
We develop a theory describing the transition to a spatially homogeneous regime in a mixing flow with a chaotic in time reaction. The transverse Lyapunov exponent governing the stability of the homogeneous state can be represented as a combination of Lyapunov exponents for spatial mixing and temporal chaos. This representation, being exact for time- independent flows and equal Peclet numbers of different components, is demonstrated to work accurately for time- dependent flows and different Peclet numbers
Ordinary differential equations (ODEs) have been studied for centuries as a means to model complex dynamical processes from the real world. Nevertheless, their application to sound synthesis has not yet been fully exploited. In this article we present a systematic approach to sound synthesis based on first-order complex and real ODEs. Using simple time-dependent and nonlinear terms, we illustrate the mapping between ODE coefficients and physically meaningful control parameters such as pitch, pitch bend, decay rate, and attack time. We reveal the connection between nonlinear coupling terms and frequency modulation, and we discuss the implications of this scheme in connection with nonlinear synthesis. The ability to excite a first-order complex ODE with an external input signal is also examined; stochastic or impulsive signals that are physically or synthetically produced can be presented as input to the system, offering additional synthesis possibilities, such as those found in excitation/filter synthesis and filter-based modal synthesis.
We investigate synchronization of coupled organ pipes. Synchronization and reflection in the organ lead to undesired weakening of the sound in special cases. Recent experiments have shown that sound interaction is highly complex and nonlinear, however, we show that two delay-coupled Van-der-Pol oscillators appear to be a good model for the occurring dynamical phenomena. Here the coupling is realized as distance-dependent, or time-delayed, equivalently. Analytically, we investigate the synchronization frequency and bifurcation scenarios which occur at the boundaries of the Arnold tongues. We successfully compare our results to experimental data.
Machine learning control
(2020)
Recently, the term explainable AI came into discussion as an approach to produce models from artificial intelligence which allow interpretation. For a long time, symbolic regression has been used to produce explainable and mathematically tractable models. In this contribution, we extend previous work on symbolic regression methods to infer the optimal control of a dynamical system given one or several optimization criteria, or cost functions. In earlier publications, network control was achieved by automated machine learning control using genetic programming. Here, we focus on the subsequent path continuation analysis of the mathematical expressions which result from the machine learning model. In particular, we use AUTO to analyze the solution properties of the controlled oscillator system which served as our model. As a result, we show that there is a considerable advantage of explainable symbolic regression models over less accessible neural networks. In particular, the roadmap of future works may be to integrate such analyses into the optimization loop itself to filter out robust solutions by construction.
We study the modeling and prediction of dynamical systems based on conventional models derived from measurements. Such algorithms are highly desirable in situations where the underlying dynamics are hard to model from physical principles or simplified models need to be found. We focus on symbolic regression methods as a part of machine learning. These algorithms are capable of learning an analytically tractable model from data, a highly valuable property. Symbolic regression methods can be considered as generalized regression methods. We investigate two particular algorithms, the so-called fast function extraction which is a generalized linear regression algorithm, and genetic programming which is a very general method. Both are able to combine functions in a certain way such that a good model for the prediction of the temporal evolution of a dynamical system can be identified. We illustrate the algorithms by finding a prediction for the evolution of a harmonic oscillator based on measurements, by detecting an arriving front in an excitable system, and as a real-world application, the prediction of solar power production based on energy production observations at a given site together with the weather forecast.
Big data have become a critically enabling component of emerging mathematical methods aimed at the automated discovery of dynamical systems, where first principles modeling may be intractable. However, in many engineering systems, abrupt changes must be rapidly characterized based on limited, incomplete, and noisy data. Many leading automated learning techniques rely on unrealistically large data sets, and it is unclear how to leverage prior knowledge effectively to re-identify a model after an abrupt change. In this work, we propose a conceptual framework to recover parsimonious models of a system in response to abrupt changes in the low-data limit. First, the abrupt change is detected by comparing the estimated Lyapunov time of the data with the model prediction. Next, we apply the sparse identification of nonlinear dynamics (SINDy) regression to update a previously identified model with the fewest changes, either by addition, deletion, or modification of existing model terms. We demonstrate this sparse model recovery on several examples for abrupt system change detection in periodic and chaotic dynamical systems. Our examples show that sparse updates to a previously identified model perform better with less data, have lower runtime complexity, and are less sensitive to noise than identifying an entirely new model. The proposed abrupt-SINDy architecture provides a new paradigm for the rapid and efficient recovery of a system model after abrupt changes.
Many previous studies have shown that the turbulent mixing layer under periodic forcing tends to adopt a lock-on state, where the major portion of the fluctuations in the flow are synchronized at the forcing frequency. The goal of this experimental study is to apply closed-loop control in order to provoke the lock-on state, using information from the flow itself. We aim to determine the range of frequencies for which the closed-loop control can establish the lock-on, and what mechanisms are contributing to the selection of a feedback frequency. In order to expand the solution space for optimal closed-loop control laws, we use the genetic programming control (CPC) framework. The best closed-loop control laws obtained by CPC are analysed along with the associated physical mechanisms in the mixing layer flow. The resulting closed-loop control significantly outperforms open-loop forcing in terms of robustness to changes in the free-stream velocities. In addition, the selection of feedback frequencies is not locked to the most amplified local mode, but rather a range of frequencies around it.
Mixing layer manipulation experiment from open-loop forcing to closed-loop machine learning control
(2015)
Many previous studies have shown that the turbulent mixing layer under periodic forcing tends to adopt a lock-on state, where the major portion of the fluctuations in the flow are synchronized at the forcing frequency. The goal of this experimental study is to apply closed-loop control in order to provoke the lock-on state, using information from the flow itself. We aim to determine the range of frequencies for which the closed-loop control can establish the lock-on, and what mechanisms are contributing to the selection of a feedback frequency. In order to expand the solution space for optimal closed-loop control laws, we use the genetic programming control (CPC) framework. The best closed-loop control laws obtained by CPC are analysed along with the associated physical mechanisms in the mixing layer flow. The resulting closed-loop control significantly outperforms open-loop forcing in terms of robustness to changes in the free-stream velocities. In addition, the selection of feedback frequencies is not locked to the most amplified local mode, but rather a range of frequencies around it.
The concept of a "flow network"-a set of nodes and links which carries one or more flows-unites many different disciplines, including pipe flow, fluid flow, electrical, chemical reaction, ecological, epidemiological, neurological, communications, transportation, financial, economic and human social networks. This Feature Paper presents a generalized maximum entropy framework to infer the state of a flow network, including its flow rates and other properties, in probabilistic form. In this method, the network uncertainty is represented by a joint probability function over its unknowns, subject to all that is known. This gives a relative entropy function which is maximized, subject to the constraints, to determine the most probable or most representative state of the network. The constraints can include "observable" constraints on various parameters, "physical" constraints such as conservation laws and frictional properties, and "graphical" constraints arising from uncertainty in the network structure itself. Since the method is probabilistic, it enables the prediction of network properties when there is insufficient information to obtain a deterministic solution. The derived framework can incorporate nonlinear constraints or nonlinear interdependencies between variables, at the cost of requiring numerical solution. The theoretical foundations of the method are first presented, followed by its application to a variety of flow networks.
Characterization and calibration of piezoelectric polymers in situ measurements of body vibrations
(2011)
Piezoelectric polymers are known for their flexibility in applications, mainly due to their bending ability, robustness, and variable sensor geometry. It is an optimal material for minimal-invasive investigations in vibrational systems, e.g., for wood, where acoustical impedance matches particularly well. Many applications may be imagined, e. g., monitoring of buildings, vehicles, machinery, alarm systems, such that our investigations may have a large impact on technology. Longitudinal piezoelectricity converts mechanical vibrations normal to the polymer-film plane into an electrical signal, and the respective piezoelectric coefficient needs to be carefully determined in dependence on the relevant material parameters. In order to evaluate efficiency and durability for piezopolymers, we use polyvinylidene fluoride and measure the piezoelectric coefficient with respect to static pressure, amplitude of the dynamically applied force, and long-term stability. A known problem is the slow relaxation of the material towards equilibrium, if the external pressure changes; here, we demonstrate how to counter this problem with careful calibration. Since our focus is on acoustical measurements, we determine accurately the frequency response curve - for acoustics probably the most important characteristic. Eventually, we show that our piezopolymer transducers can be used as a calibrated acoustical sensors for body vibration measurements on a wooden musical instrument, where it is important to perform minimal-invasive measurements. A comparison with the simultaneously recorded airborne sound yields important insight of the mechanism of sound radiation in comparison with the sound propagating in the material. This is especially important for transient signals, where not only the long-living eigenmodes contribute to the sound radiation. Our analyses support that piezopolymer sensors can be employed as a general tool for the determination of the internal dynamics of vibrating systems.