Real-Time Strain Field Prediction of Steel Cross Girder Based on Proper Orthogonal Decomposition (POD) and CNN-LSTM

The push-plate kiln is a kind of kiln equipment widely used in the oxygen-free sintering of high-temperature alloy materials. Its flow field monitoring has an important application value for the manufacturing industry. However, traditional simulation methods cannot meet the requirements of real-time applications due to the high computational cost and being time-consuming. The rapid development of artificial intelligence technology will empower the traditional manufacturing industry. In this paper, we propose a data-driven digital twin framework for real-time flow field prediction by combining the CFD modeling simulation, IoT, and deep learning technologies. The framework integrates geometric, rule, physical, and neural network models to achieve the real-time simulation of physical and twin objects. The proper orthogonal decomposition (POD) and multiscale convolutional neural network (MCNN) are innovatively embedded into the framework. The POD is used to map high-dimensional data to low-dimensional features, and the MCNN is used to construct models predicting low-dimensional features for fast flow field prediction. The effectiveness of the proposed model is verified by the push-plate kiln case. The results show that the digital twin can quickly predict multi-physics fields based on the perceptual data to achieve the real-time evaluation of the operating state of the push-plate kiln.

The method of proper orthogonal mode decomposition (POD) or KarhunenLoeve decomposition (KLD) is a means of extracting spatial information from a set of time-series data available from a set of sensing locations over a domain. The POD can be used to obtain low-dimensional models or discrete or distributed dynamical systems by computing an orthogonal set of eigen-functions through a finite-dimensional eigenvalue problem that is obtained by post processing of time-series measurements at different spatial locations. Interestingly enough, these eigenfunctions form an orthogonal basis (irrespective of the linear or nonlinear nature of the measured signals) which is optimal in the sense that fewer POD modes are needed to capture a given amount of energy of the measured signal than any other linear set of modes, including vibration modes [219]. Moreover, the eigenvalue corresponding to a given eigenfunction quantifies the amount of energy of the measured signal that is captured by the specific POD mode. Hence, the POD method not only provides a linear orthogonal basis of modes, but also a quantitative measure of the relative importance of these modes with regard to the energy of the signal captured by the POD analysis. This feature of the method makes it a valuable tool in the analysis, system identification and order reduction of the dynamics of engineering systems. As pointed by Kerschen [102] the POD analysis resembles the Singular Value Decomposition Method, with the later method providing additional information related to amplitude modulations of the identified waveforms.

Analysis of PIV measurements using modal decomposition techniques, POD and DMD, to study flow structures and their dynamics within a stirred-tank reactor

In this paper, a frequently employed technique named the sparsity-promoting dynamic mode decomposition (SPDMD) is proposed to analyze the velocity fields of atmospheric motion. The dynamic mode decomposition method (DMD) is an effective technique to extract dynamic information from flow fields that is generated from direct experiment measurements or numerical simulation and has been broadly employed to study the dynamics of the flow, to achieve a reduced-order model (ROM) of the complex high dimensional flow field, and even to predict the evolution of the flow in a short time in the future. However, for standard DMD, it is hard to determine which modes are the most physically relevant, unlike the proper orthogonal decomposition (POD) method which ranks the decomposed modes according to their energy content. The advanced modal decomposition method SPDMD is a variant of the standard DMD, which is capable of determining the modes that can be used to achieve a high-quality approximation of the given field. It is novel to introduce the SPDMD to analyze the atmospheric flow field. In this study, SPDMD is applied to extract essential dynamic information from the 200 hPa jet flow, and the decomposed results are compared with the POD method. To further demonstrate the extraction effect of POD/SPDMD methods on the 200 hPa jet flow characteristics, the POD/SPDMD reduced-order models are constructed, respectively. The results show that both modal decomposition methods successfully extract the underlying coherent structures from the 200 hPa jet flow. And the DMD method provides additional information on the modal properties, such as temporal frequency and growth rate of each mode which can be used to identify the stability of the modes. It is also found that a fewer order of modes determined by the SPDMD method can capture nearly the same dynamic information of the jet flow as the POD method. Furthermore, from the quantitative comparisons between the POD and SPDMD reduced-order models, the latter provides a higher precision than the former, especially when the number of modes is small.

The analysis of the unsteady flow field in axial compressor cascade is conducted using methods such as proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). Data on the unsteady flow field of the Stage-35 compressor cascade are acquired via computational fluid dynamics (CFD) simulations and subsequently processed using POD and DMD for dimensionality reduction. Using singular value decomposition, the POD technique identifies the dominant modes, showing that the first nine modes account for 99% of the energy in the flow field, thus highlighting the primary flow structures. On the other hand, the DMD approach isolates the periodic and high-frequency dynamics within the flow field by decomposing the dynamic modes, effectively identifying fine variations in the unsteady flow. The study examines the flow field at three distinct moments within an unsteady cycle, specifically at 1/4T, 1/2T, and 3/4T, reconstructing the flow field at each instance and performing root mean square error analysis. Reconstruction results and error analysis demonstrate that the POD method excels at reconstructing low-frequency features, whereas the DMD method accurately identifies the unsteady dynamic aspects of the flow field, excelling in resolving high-frequency details. Both methods demonstrate high feasibility regarding the accuracy and efficiency of flow field reconstruction.

Due to a lack of environment awareness, today's low-altitude fixed-wing Unmanned Aerial Vehicles (UAVs) are limited to primitively follow user-defined waypoints. All high-level decision making is still performed by a human user. Fully-autonomous remote missions in complex environments however require true environment awareness both with respect to terrain and wind. While terrain-aware navigation is covered in existing literature, the real-time estimation and consideration of complex wind patterns onboard UAVs is not. This paper therefore presents the literature's first-ever local 3D wind field prediction method which can run in real time onboard a UAV. The selected method is a simple downscaling approach which retrieves low-resolution data from global weather models and then adjusts the wind field via potential flow theory such that terrain boundaries, mass conservation, and the atmospheric stratification are observed. Typical 3D wind fields of 1 km <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> volume are calculated in below 10 seconds. Synthetic test cases such as the flow around a semi-cylinder, through a valley and over a ramp yield good results. A comparison with 3D LIDAR wind data collected over 10 days in the Swiss Alps shows an overall wind error reduction of 23% with respect to the zero-wind assumption that is mostly used for UAV path planning today. Overall, our initial research demonstrates the feasibility of real-time 3D wind field prediction onboard a UAV. However, the focus on low computation time means that the vertical wind prediction lacks accuracy. The paper therefore ends with a research outlook into real-time 3D wind field prediction through the fusion of machine learning techniques with Computational Fluid Dynamics (CFD) methods.

A comparison of empirical orthogonal decomposition methods in baroclinic flows

High-resolution solar observations show the complex structure of the magnetohydrodynamic (MHD) wave motion. We apply the techniques of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) to identify the dominant MHD wave modes in a sunspot using the intensity time series. The POD technique was used to find modes that are spatially orthogonal, whereas the DMD technique identifies temporal orthogonality. Here, we show that the combined POD and DMD approaches can successfully identify both sausage and kink modes in a sunspot umbra with an approximately circular cross-sectional shape. This article is part of the Theo Murphy meeting issue 'High-resolution wave dynamics in the lower solar atmosphere'.

Modal Analysis of Fluid Flows: An Overview

This study analyzed the pressure patterns and local pressure of tall buildings with corner modifications (recessed and chamfered corner) using wind tunnel tests and proper orthogonal decomposition (POD). POD can distinguish pressure patterns by POD mode and more dominant pressure patterns can be found according to the order of POD modes. Results show that both recessed and chamfered corners effectively reduced wind-induced responses. Additionally, unique effects were observed depending on the ratio of corner modification. Tall building models with recessed corners showed fluctuations in the approaching wind flow in the first POD mode and vortex shedding effects in the second POD mode. With large corner modification, energy distribution became small in the first POD mode, which shows that the effect of the first POD mode reduced. Among building models with chamfered corners, vortex shedding effects appeared in the first POD mode, except for the model with the highest ratio of corner modifications. The POD confirmed that both recessed and chamfered corners play a role in reducing vortex shedding effects, and the normalized power spectral density peak value of modes showing vortex shedding was smaller than that of the building model with a square section. Vortex shedding effects were observed on the front corner surfaces resulting from corner modification, as with the side surface. For buildings with recessed corners, the local pressure on corner surfaces was larger than that of side surfaces. Moreover, the average wind pressure was effectively reduced to 88.42% and 92.40% in RE1 on the windward surface and CH1 on the side surface, respectively.

This paper considers use of Proper Orthogonal Decomposition (POD), also known as the Karhunen-Loeve (K-L) method, to obtain reduced order dynamic models of nonlinear structural systems. The study applies POD to simulated time series data to extract dominant “modes” that describe the system behavior. The “POD modes” are used to formulate reduced order differential equation models (ROM’s) of the structure in which the dependent variables are the POD modal coordinates. Two example systems are considered: 1) a clamped beam whose tip is placed between attracting magnets; POD analysis of this system was done by Feeny and Kerschen [1] using harmonic excitation to excite chaotic motions, which were analyzed to develop the POD modes for reduced order modeling, and 2) a chain of oscillators having an isolated nonlinear Duffing element. Our approach is to generate the POD modes for model reduction by using strong, band limited random excitation to excite vibratory motions. The richness of this type of excitation is intended to provide responses whose POD-based reduced models can be used with reasonable accuracy for system parameters that differ from those used to generate the reduced order model (this is the central issue in using POD to generate reduced order nonlinear models of structures). Our results indicate that, due to the spectral richness of the random excitation, this type of excitation can be used with reasonable accuracy for conditions that differ from those used to generate the reduced order model. The method works well for the chain of oscillator system, but less well for the magnetic beam system, due to the presence of multiple stable equilibria in this system. A useful result of this work is the finding that the number of POD modes required to achieve accurate reduced order differential equation models may be considerably larger than the number of POD modes needed to accurately project the full model responses onto a subspace defined by dominant POD modes. Also, it is shown that for the clamped beam problem with multiple equilibria, we require more modes to develop a reduced order model than we require for a chain of oscillators.

We apply a simple proper orthogonal decomposition (POD) method to compute the nonlinear oscillations of a degenerate two-dimensional fluttering plate undergoing supersonic flow. First, the von Karman's large deflection theory and quasi-steady aerodynamic theory are employed in constructing the governing equations of the simply supported plate. Then, the governing equations are solved by both the Galerkin method and the POD method. The Galerkin method is ac- curate but sometimes computationally expensive, since the number of degrees of freedom (dofs) required is relatively large provided that nonlinearity is strong. The POD method can be used to capture the complex dynamics of a strongly nonlinear system using very few degrees of freedom, much fewer than the Galerkin approach. The presently proposed POD method has two advantages over the conventional one. i) a simple numerical difference technique is first introduced to the POD method to avoid the complicated mode-to-mode projection between POD modes and Galerkin modes. ii) POD based reduced order models (POD/ROM) are constructed by us- ing a set of general modes which is extracted from chaotic responses. That is to say POD modes extracted from one set of parameters can be applied to various parameter variations for the same dynamic system. Moveover, results for the buck- led, LCO and chaotic responses of the plate are presented and compared with the Galerkin solutions. Numerical examples demonstrate the accuracy and efficiency of the present POD method.

Many industrial chemical processes are complex, multi-phase and large scale in nature. These processes are characterized by various nonlinear physiochemical effects and fluid flows. Such processes often show coexistence of fast and slow dynamics during their time evolutions. The increasing demand for a flexible operation of a complex process, a pressing need to improve the product quality, an increasing energy cost and tightening environmental regulations make it rewarding to automate a large scale manufacturing process. Mathematical tools used for process modeling, simulation and control are useful to meet these challenges. Towards this purpose, development of process models, either from the first principles (conservation laws) i.e. the rigorous models or the input-output data based models constitute an important step. Both types of models have their own advantages and pitfalls. Rigorous process models can approximate the process behavior reasonably well. The ability to extrapolate the rigorous process models and the physical interpretation of their states make them more attractive for the automation purpose over the input-output data based identified models. Therefore, the use of rigorous process models and rigorous model based predictive control (R-MPC) for the purpose of online control and optimization of a process is very promising. However, due to several limitations e.g. slow computation speed and the high modeling efforts, it becomes difficult to employ the rigorous models in practise. This thesis work aims to develop a methodology which will result in smaller, less complex and computationally efficient process models from the rigorous process models which can be used in real time for online control and dynamic optimization of the industrial processes. Such methodology is commonly referred to as a methodology of Model (order) Reduction. Model order reduction aims at removing the model redundancy from the rigorous process models. The model order reduction methods that are investigated in this thesis, are applied to two benchmark examples, an industrial glass manufacturing process and a tubular reactor. The complex, nonlinear, multi-phase fluid flow that is observed in a glass manufacturing process offers multiple challenges to any model reduction technique. Often, the rigorous first principle models of these benchmark examples are implemented in a discretized form of partial differential equations and their solutions are computed using the Computational Fluid Dynamics (CFD) numerical tools. Although these models are reliable representations of the underlying process, computation of their dynamic solutions require a significant computation efforts in the form of CPU power and simulation time. The glass manufacturing process involves a large furnace whose walls wear out due to the high process temperature and aggressive nature of the molten glass. It is shown here that the wearing of a glass furnace walls result in change of flow patterns of the molten glass inside the furnace. Therefore it is also desired from the reduced order model to approximate the process behavior under the influence of changes in the process parameters. In this thesis the problem of change in flow patterns as result of changes in the geometric parameter is treated as a bifurcation phenomenon. Such bifurcations exhibited by the full order model are detected using a novel framework of reduced order models and hybrid detection mechanisms. The reduced order models are obtained using the methods explained in the subsequent paragraphs. The model reduction techniques investigated in this thesis are based on the concept of Proper Orthogonal Decompositions (POD) of the process measurements or the simulation data. The POD method of model reduction involves spectral decomposition of system solutions and results into arranging the spatio-temporal data in an order of increasing importance. The spectral decomposition results into spatial and temporal patterns. Spatial patterns are often known as POD basis while the temporal patterns are known as the POD modal coefficients. Dominant spatio-temporal patterns are then chosen to construct the most relevant lower dimensional subspace. The subsequent step involves a Galerkin projection of the governing equations of a full order first principle model on the resulting lower dimensional subspace. This thesis can be viewed as a contribution towards developing the databased nonlinear model reduction technique for large scale processes. The major contribution of this thesis is presented in the form of two novel identification based approaches to model order reduction. The methods proposed here are based on the state information of a full order model and result into linear and nonlinear reduced order models. Similar to the POD method explained in the previous paragraph, the first step of the proposed identification based methods involve spectral decomposition. The second step is different and does not involve the Galerkin projection of the equation residuals. Instead, the second step involves identification of reduced order models to approximate the evolution of POD modal coefficients. Towards this purpose, two different methods are presented. The first method involves identification of locally valid linear models to represent the dynamic behavior of the modal coefficients. Global behavior is then represented by ‘blending’ the local models. The second method involves direct identification of the nonlinear models to represent dynamic evolution of the model coefficients. In the first proposed model reduction method, the POD modal coefficients, are treated as outputs of an unknown reduced order model that is to be identified. Using the tools from the field of system identification, a blackbox reduced order model is then identified as a linear map between the plant inputs and the modal coefficients. Using this method, multiple local reduced LTI models corresponding to various working points of the process are identified. The working points cover the nonlinear operation range of the process which describes the global process behavior. These reduced LTI models are then blended into a single Reduced Order-Linear Parameter Varying (ROLPV) model. The weighted blending is based on nonlinear splines whose coefficients are estimated using the state information of the full order model. Along with the process nonlinearity, the nonlinearity arising due to the wear of the furnace wall is also approximated using the RO-LPV modeling framework. The second model reduction method that is proposed in this thesis allows approximation of a full order nonlinear model by various (linear or nonlinear) model structures. It is observed in this thesis, that, for certain class of full order models, the POD modal coefficients can be viewed as the states of the reduced order model. This knowledge is further used to approximate the dynamic behavior of the POD modal coefficients. In particular, reduced order nonlinear models in the form of tensorial (multi-variable polynomial) systems are identified. In the view of these nonlinear tensorial models, the stability and dissipativity of these models is investigated. During the identification of the reduced order models, the physical interpretation of the states of the full order rigorous model is preserved. Due to the smaller dimension and the reduced complexity, the reduced order models are computationally very efficient. The smaller computation time allows them to be used for online control and optimization of the process plant. The possibility of inferring reduced order models from the state information of a full order model alone i.e. the possibility to infer the reduced order models in the absence of access to the governing equations of a full order model (as observed for many commercial software packages) make the methods presented here attractive. The resulting reduced order models need further system theoretic analysis in order to estimate the model quality with respect to their usage in an online controller setting.

The turbulent properties of a heated and unheated Mach 0.85 axisymmetric jet have been studied. The velocity field of the jet at static temperature ratios of 0.87 and 2.34, was measured in the streamwise radial plane using Particle Image Velocimetry. The velocity measurements were acquired between streamwise locations of 3D and 8D downstream from the nozzle exit. Proper Orthogonal Decomposition (POD) was applied to the velocity field using snapshot POD. The POD analysis showed that the eigenvalues of the heated jet had higher fraction of energy. The POD eigenfunctions or modes of the streamwise velocity of both jets were similar, while the POD modes of the radial velocity of both jets were very different. The POD modes of radial velocity of the unheated jet were symmetrical about the jet centerline, and the modes of the heated jet seemed to merge at the centerline. The near field pressure measurements were acquired just outside the shearlayer. A linear array of five pressure transducers was placed at 7° to the nozzle lipline, so that it would be parallel to the shear layer. The transducers in the array were spaced one diameter apart. Pressure measurements were acquired at streamwise locations between 4.25D and 10.25D from the nozzle exit. Based on the slope of the pressure spectra, the propagating events in the two jets were identified. The POD was also applied to the pressure data, and the POD modes of the two jets were compared. The peak in the amplitude of the POD mode of the heated jet was at a higher frequency. With increasing mode numbers, the peak in the POD mode of both jets shifted to a downstream location.

The cycle-to-cycle variation of engine in-cylinder flow is critical for the improvement of performance for spark-ignition internal combustion engines. Proper orthogonal decomposition (POD), with its ability to extract the most energetic fluctuation structure, is widely used to analyze the in-cylinder flow and understand the variation of its evolution in different cycles. However, both of the two existing approaches to use POD for engine flow analysis encounter difficulties when applied for this purpose. Phase-dependent POD decomposes a data set in which all samples are taken at a certain engine phase (crank angle) from different cycles, but the POD results at neighboring engine phases do not necessarily evolve coherently. Phase-invariant POD, when applied to analyze tumble flow, stretches/compresses and interpolates the flow fields obtained at different engine phases onto the same grid, and this deformation means that phase-invariant POD results are no longer significant in energy sense. To overcome these difficulties, we propose an adaptation of conditional space-time POD to work with engine flow, with which the flow within a range of engine phases in each cycle is considered as one sample. It is shown that the low-order modes obtained with conditional space-time POD capture fluctuation structures that evolve coherently, and these results are compared and contrasted with those of the two existing POD approaches. A reduced-order model of the engine in-cylinder flow is constructed based on the partial sum of the modes and coefficients obtained from the conditional space-time POD, and it is shown that this new reduced-order model identifies structure that is both coherent spatially and temporally.