Orthogonal Decomposition Research Articles

Image Retrievable Encryption Based on Linear Fitting and Orthogonal Transformation

Abstract With the development of cloud computing, an increasing number of resource-constrained image owners tend to store their images in the cloud and rely on image retrieval services to obtain the images they desire. However, the security of the cloud cannot be fully guaranteed. To ensure image security while achieving good retrieval performance, we have designed a retrievable image encryption algorithm based on linear fitting and orthogonal transformation. This algorithm first generates encryption and feature extraction domains through orthogonal decomposition, and then applies a modified ResNet50 network for feature extraction in the feature extraction domain. The encryption process employs an improved affine transformation based on linear fitting, where part of the fitting values comes from the original image data and the other part comes from data generated by a chaotic system. Additionally, to simplify the measurement of feature similarity in the cloud, we have designed a hierarchical feature index tree to narrow the retrieval scope, thereby reducing retrieval complexity. Experimental results show that the proposed algorithm effectively protects image privacy and achieves high retrieval accuracy. The F-score reached 6.7634% on the Ghim10k dataset and 25.514% on the Corel 1K dataset, significantly improving upon traditional methods. This algorithm has potential application value in the fields of secure image storage and efficient retrieval in the cloud.

Application of the L2-orthogonal decomposition of the second fundamental form of the hypersurfaces and the ahlfors laplacian to the study of the general relativistic vacuum constraint equations

Application of the <i>L</i>²-orthogonal decomposition of the second fundamental form of the hypersurfaces and the ahlfors laplacian to the study of the general relativistic vacuum constraint equations

Effect of rotation on wake vortices in stratified flow

Stratified wakes past an isolated conical seamount are simulated at a Froude number of $Fr = 0.15$ and Rossby numbers of $Ro = 0.15$ , 0.75 and $\infty$ . The wakes exhibit a Kármán vortex street, unlike their unstratified, non-rotating counterpart. Vortex structures are studied in terms of large-scale global modes, as well as spatially localised vortex evolution, with a focus on rotation effects. The global modes are extracted by spectral proper orthogonal decomposition (SPOD). For all three studied $Ro$ ranging from mesoscale, submesoscale and non-rotating cases, the frequency of the SPOD modes at different heights remains coupled as a global constant. However, the shape of the SPOD modes changes from slanted ‘tongues’ at zero rotation ( $Ro=\infty$ ) to tall hill-height columns at strong rotation ( $Ro=0.15$ ). A novel method for vortex centre tracking shows that, in all three cases, the vortices at different heights advect uniformly at approximately $0.9U_{\infty }$ beyond the near wake, consistent with the lack of variability of the global modes. Under system rotation, cyclonic vortices and anticyclonic vortices (AVs) present considerable asymmetry, especially at $Ro = 0.75$ . The vorticity distribution as well as the stability of AVs are tracked downstream using statistics conditioned to the identified vortex centres. At $Ro=0.75$ , intense AVs with relative vorticity up to $\omega _z/f_{c}=-2.4$ (where $\omega_z$ is the vertical vorticity and $f_c$ is the Coriolis frequency) are seen with small regions of instability and they maintain large $\omega _z/f_{c}$ magnitude in the far wake. Recent stability analysis that accounts for stratification and viscosity is found to improve on earlier criteria and show that these intense AVs are stable.

Nonlinear multi-image optical authentication based on QR decomposition and Kramer-Kronig relations

Abstract In this paper, a new nonlinear optical multi-image authentication scheme is proposed based on Kramers-Kronig digital holography and orthogonal triangular decomposition or QR decomposition. Here, the complex light field carrying the information of multiple images is modulated by random phase masks and propagated at certain distance. Afterwards, the QR decomposition is applied to the complex wavefront to generate the private keys and to add the non-linearity in the scheme. Next, the product of orthogonal matrix and upper triangular matrix is processed further. The obtained output is modulated by different phase masks and interfered with reference beam to record the encrypted image. For decryption, the Kramer-Kronig relation is utilized to extract the plaintext images directly with only the positive frequency part. A series of numerical simulations are conducted to validate the efficacy and robustness of proposed image authentication scheme.&amp;#xD;

A non-intrusive reduced-order model for finite element analysis of implant positioning in total hip replacements.

Sophisticated high-fidelity simulations can predict bone mass density (BMD) changes around a hip implant after implantation. However, these models currently have high computational demands, rendering them impractical for clinical settings. Model order reduction techniques offer a remedy by enabling fast evaluations. In this work, a non-intrusive reduced-order model, combining proper orthogonal decomposition with radial basis function interpolation (POD-RBF), is established to predict BMD distributions for varying implant positions. A parameterised finite element mesh is morphed using Laplace's equation, which eliminates tedious remeshing and projection of the BMD results on a common mesh in the offline stage. In the online stage, the surrogate model can predict BMD distributions for new implant positions and the results are visualised on the parameterised reference mesh. The computational time for evaluating the final BMD distribution around a new implant position is reduced from minutes to milliseconds by the surrogate model compared to the high-fidelity model. The snapshot data, the surrogate model parameters and the accuracy of the surrogate model are analysed. The presented non-intrusive surrogate model paves the way for on-the-fly evaluations in clinical practice, offering a promising tool for planning and monitoring of total hip replacements.

A synchronization method based on current orthogonal decomposition considering harmonic components for wireless power transfer system

AbstractIn order to improve the wireless power transfer (WPT) system, an active rectifier is commonly used. Synchronization technology is critical for active rectifier. The traditional synchronization method based on zero‐crossing detection (ZCD) is unstable and not suitable for the case with lots of harmonic component in the resonant current. Here, a new synchronization method with harmonic compensation is proposed for the WPT system without the ZCD of the resonant current. A mathematical model is built to analyse and compensate the impact caused by the harmonic on phase angle. Therefore, the proposed method can be applied to the high‐order resonant network like LCC compensated topology where the harmonic component is rich in the resonant current. For verifying the proposed synchronization method, a laboratory prototype is built. The experimental results prove that the system can operate stably with parameter perturbation and different load. The peak transfer efficiency of the system can achieve the 94.5%.

Noise Source Identification for the Unsteady Low-Pressure Turbine Flow Fields

Abstract The article presents a data processing methodology to identify the noise sources in low-pressure turbine (LPT) stages and their relative importance. Scale adaptive simulation (SAS) has been carried out on a geometry reproducing the midspan blade section of an entire LPT stage. The proper orthogonal decomposition (POD) is applied to data matrices constructed with the velocity and the pressure fields in order to distinctly extract the coherent structures responsible for velocity fluctuations and pressure waves (i.e., POD modes from the corresponding kernel), their temporal evolution and energies. The cross-correlation matrices of the pressure modes and the velocity modes reveal the degree of coupling between pressure and velocity oscillations, thus highlighting the modes linked to the same flow phenomena. The results show that the periodic vortex shedding at the stator trailing edge emits strong pressure waves, and the upstream-propagating waves reflect against the adjacent stator suction surface. The POD modes related to the rotor–stator interaction show peak amplitudes at the blade passing frequency and its harmonics and their shapes mimic the potential interaction in the stator domain and the migration of viscous wakes in the rotor domain. The POD modes with lower energy content correspond to the stochastic turbulent structures originated from the turbulent boundary layers as well as those carried by the vane and blade wakes. The biggest noise sources of the current flow fields are identified to be the von Karman vortex street downstream the stator trailing edge and the rotor–stator interaction.

Deep learning based hybrid POD-LSTM framework for laminar natural convection flow in a rectangular enclosure

Abstract Laminar natural convection in side-heated enclosures is characterized by transient phenomena of the working fluid till it reaches steady state. The side heating can done in several ways the most common way being heating one end at constant temperature and cooling the other end. One of the other ways is heating both sides of the enclosure at a constant heat flux. Mathematical modeling of such problems using Computational Fluid Dynamics (CFD) essentially involves considerable amount of computational time and power to predict the flow phenomena observed in actual experimentation. In the last few years, data driven model frameworks have proven to be anefficient way in saving both time and computational cost in several applications. In the present study, a data driven model framework using a combination of unsupervised machine learning (using Proper Orthogonal Decomposition [POD]) and supervised deep learning models (using Long Short Term Memory [LSTM]) has been developed and referred to as POD-LSTM framework. The selection of a few dominant spatial bases and accompanying temporal modes provides us with a reduced order model of the system. The flow is then reconstructed and compared with results of CFD simulations. The Rayleigh number (Ra) chosen for the study is 3.27 × 1010. The estimated time to reach stedy state for this Ra number is 15,000 s. The POD-LSTM framework is trained using data obtained from a validated CFD model for the first 1,000 s. The trained model was then tested to predict temporal dynamics for the entire 15,000 s. The predictions provided by POD-LSTM framework were found upto 98 % accurate compared to the ones predicted by CFD. The computational time and power was however an order of magnitude lower for the POD-LSTM framework than that required for the CFD model.

Prediction of Temperature Distribution on an Aircraft Hot-Air Anti-Icing Surface by ROM and Neural Networks

To address the inefficiencies and time-consuming nature of traditional hot-air anti-icing system designs, reduced-order models (ROMs) and machine learning techniques are introduced to predict anti-icing surface temperature distributions. Two models, AlexNet combined with Proper Orthogonal Decomposition (POD-AlexNet) and multi-CNNs with GRU (MCG), are proposed by comparing several classic neural networks. Design variables of the hot-air anti-icing cavity are used as inputs of the two models, and the corresponding surface temperature distribution data serve as outputs, and then the performance of these models is evaluated on the test set. The POD-AlexNet model achieves a mean prediction accuracy of over 95%, while the MCG model reaches 96.97%. Furthermore, the proposed model demonstrates a prediction time of no more than 5.5 ms for individual temperature samples. The proposed models not only provide faster predictions of anti-icing surface temperature distributions than traditional numerical simulation methods but also ensure acceptable accuracy, which supports the design of aircraft hot-air anti-icing systems based on optimization methods such as genetic algorithms.

Enhancing data representation in forging processes: Investigating discretization and R-adaptivity strategies with Proper Orthogonal Decomposition reduction

Effective data reduction techniques are crucial for enhancing computational efficiency in complex industrial processes such as forging. In this study, we investigate various discretization and mesh adaptivity strategies using Proper Orthogonal Decomposition (POD) to optimize data reduction fidelity in forging simulations. We focus particularly on r-adaptivity techniques, which ensure a consistent number of elements throughout the field representation, filling a gap in existing research that predominantly concentrates on h-adaptivity. Our investigation compares isotropic mesh approaches with anisotropic mesh adaptations, including gradient-based, isolines-based, and spring-energy-based methods. Through numerical simulations and analysis, we demonstrate that these anisotropic techniques provide superior fidelity in representing deformation fields compared to isotropic meshes. These improvements are achieved while maintaining a similar level of model reduction efficiency. This enhancement in representation leads to improved data reduction quality, forming the foundation for data-driven models. This research contributes to advancing the understanding of mesh adaptivity approaches and their potential applications in data-driven modeling across various industrial domains.

Constraint‐Driven Multilane Merging Control for Underactuated Connected and Automated Vehicle System With Mismatched Uncertainty

ABSTRACTThis article investigates the merging control for underactuated connected and automated vehicle (CAV) systems with mismatched uncertainty. The objective is to guarantee the safe and prescribed dynamic performance of the system during multilane merging. For CAV multidimensional motions, a strongly coupled vehicle dynamics with time‐varying uncertainties is constructed. For the nominal system, the control objectives are designed as system constraints, with equality constraints guaranteeing merging and time‐varying inequality constraints guaranteeing dynamic performance bounds. Based on the diffeomorphism method and bounded constraint, the system constraints are reconstructed to a unified representation. Combining system constraints and underactuated structure, the nominal controller is derived based on constraint‐following. For the uncertain system, the uncertainty is decomposed into matched and mismatched portions by orthogonal decomposition. The adaptive robust controller is designed based only on matched uncertainty. Through Lyapunov minimax analysis, the control renders the errors uniformly bounded and uniformly ultimately bounded. Simulation results show that merging and platooning are effectively achieved with the proposed control. The system has excellent transient and steady state performance even in the presence of time‐varying uncertainties.

A Preliminary Study on The Causes of the "Triple-peak" La Niña During 2020-2022

The 2020-2022 triple-peak La Niña is the first consecutive three-year La Niña event in this century. In order to investigate the similarities and differences between this event and the other observed bimodal and triple events, this study uses synthetic and regression analysis, empirical orthogonal function decomposition, and mixed layer heat budget diagnosis to compare other events with this one in terms of location of anomalous negative sea surface temperature (SST), anomalous wind direction, and mixed layer heat budget. In the previous bimodal events, the centers of SST negative anomalies were both in the (5°S-5°N, 180°-120°W) region; there were anomalous northerly and westerly winds east of 140°W, and the negat precipitation anomalies were distributed along north of the equator. In contrast, the peak position of the second negative SST anomaly in this event is more eastward and southward; the anomalous easterlies continue from 120°E to the eastern boundary of the basin, and there is an anomalous southerlies in the eastern part of the basin. To further investigate the role of the anomalous easterlies and southerlies, this study calculates the mixed layer heat budget in key region. Corresponding to the latitudinal (longitudinal) wind anomalies, the latitudinal (longitudinal) temperature advection dominates the robust difference between the bimodal event and this triple-peak event. The anomalous easterlies and southerlies in the equatorial Pacific are favorable to the appearance of the third La Niña year in this event. This study is important in understanding the dynamics of triple-peak La Niña events, and will facilitate the forecast of such event.

Reduced basis method for the elastic scattering by multiple shape-parametric open arcs in two dimensions

We consider the elastic scattering problem by multiple disjoint arcs or cracks in two spatial dimensions. A key aspect of our approach lies in the parametric description of each arc's shape, which is controlled by a potentially high-dimensional, possibly countably infinite, set of parameters. We are interested in the efficient approximation of the parameter-to-solution map employing model order reduction techniques, specifically the reduced basis method. Initially, we utilize boundary potentials to transform the boundary value problem, originally posed in an unbounded domain, into a system of boundary integral equations set on the parametrically defined open arcs. Our aim is to construct a rapid surrogate for solving this problem. To achieve this, we adopt the two-phase paradigm of the reduced basis method. In the offline phase, we compute solutions for this problem under the assumption of complete decoupling among arcs for various shapes. Leveraging these high-fidelity solutions and Proper Orthogonal Decomposition (POD), we construct a reduced-order basis tailored to the single arc problem. Subsequently, in the online phase, when computing solutions for the multiple arc problem with a new parametric input, we utilize the aforementioned basis for each individual arc. To expedite the offline phase, we employ a modified version of the Empirical Interpolation Method (EIM) to compute a precise and cost-effective affine representation of the interaction terms between arcs. Finally, we present a series of numerical experiments demonstrating the advantages of our proposed method in terms of both accuracy and computational efficiency.

The structure and dynamics of the laminar separation bubble

A novel selective mode decomposition, proper orthogonal decomposition and dynamic mode decomposition methods are used to analyse large-eddy simulation data of the flow field about a NACA0012 airfoil at low Reynolds numbers of $5\times 10^4$ and $9\times 10^4$ , and at near-stall conditions. The objective of the analysis is to investigate the structure of the laminar separation bubble (LSB) and its associated low-frequency flow oscillation (LFO). It is shown that the flow field can be decomposed into three dominant flow modes: two low-frequency modes (LFO-Mode-1 and LFO-Mode-2) that govern an interplay of a triad of vortices and sustain the LFO phenomenon, and a high-frequency oscillating (HFO) mode featuring travelling Kelvin–Helmholtz waves along the wake of the airfoil. The structure and dynamics of the LSB depend on the energy content of these three dominant flow modes. At angles of attack lower than the stall angle of attack and above the angle of a full stall, the flow is dominated by the HFO mode. At angles of attack above the stall angle of attack the LFO-Mode-2 overtakes the HFO mode, triggers instability in the LSB and initiates the LFO phenomenon. Previous studies peg the structure, stability and bursting conditions of the separation bubble to local flow parameters. However, the amplitude of these local flow parameters is dependent on the energy content of the three dominant flow modes. Thus, the present work proposes a more robust bursting criterion that is based on global eigenmodes.

A novel dimension reduction model based on POD and two-grid Crank–Nicolson mixed finite element methods for 3D nonlinear elastodynamic sine–Gordon problem

Earthquake, petroleum and gas extraction, and underground nuclear test all could cause nonlinear tectonic deformation. It is of great significance to effectively predict and evaluate the disasters and impacts of these geological structure deformation. In this end, a three-dimension (3D) nonlinear elastodynamic sine–Gordon model that can be used to describe nonlinear tectonic deformation including 3D displacement vector, 3 × 3 symmetric stress tensor matrix, nonlinear sin term, and singular initial value functions is first proposed. Then, a new time semi-discrete mixed Crank–Nicolson (TSDMCN) scheme for the 3D nonlinear elastodynamic sine–Gordon model is developed, and the existence, stability, and error estimates of the TSDMCN solutions are proved. Next, a new two-grid mixed finite element Crank–Nicolson (TGMFECN) method with unconditional stability for the 3D nonlinear elastodynamic sine–Gordon model is developed, and the existence, stability, and error estimates of the TGMFECN solutions are proved. Thenceforth, it is the most important thing is that a novel reduced-dimensional iterative TGMFECN (RDITGMFECN) method in matrix form is established by resorting to proper orthogonal decomposition only to lower the unknown TGMFECN solution coefficient vectors and keep TGMFECN basis functions unchanged, which can ensure that the RDITGMFECN method has the same accuracy as the usual TGMFECN method, but can greatly lower the dimension of the unknown TGMFECN solution coefficient vectors so as to mitigate calculated workload, save CPU operating-time, improve computing efficiency, and improve real-time calculating accuracy. In theory, the existence, stability, and error estimates of RDITGMFECN solutions are demonstrated by matrix analysis such that the theoretical analysis becomes very intuitive and easy to be understood by the public, which is a new attempt of theoretical analysis. In application, two numerical examples are used to simulate the 3D nonlinear tectonic deformation caused by earthquake and to verify the correctness of our theoretical results and the effectiveness of the RDITGMFECN method.

Experimental study on the aerodynamic characteristics and wake flow feature of triangular prisms with various round corner radius

In this work, particle image velocimetry (PIV) and wind pressure measurements are employed in the wind tunnel to investigate triangular prisms with different rounded corner radius. The effect of rounded corner radius on wind pressure distribution, aerodynamic force, and wake flow characteristics are analyzed. Results indicate that the surface wind pressure can be effectively reduced through filleting treatment, with an average drag coefficient decrease of 52.7% as the corner shape transitions from sharp to round. The continuous increase of the radius of the rounded corner causes an accretion of the fluctuating lift coefficient, and a slow decrease of the drag coefficient. The modification of the corner radius has a limited impact on the vortex shedding frequency, but the shedding vortex tend to gather the leeward side. Additionally, statistical analysis of wake flow shows that an increment in the rounded corner radius leads to a significant reduction in the wake recirculation region, amounting to 20.16%. Concurrently, the peak values of turbulent kinetic energy and Reynolds stress exhibit a pattern of initial increase followed by a decrease. On the other hand, results of proper orthogonal decomposition suggest that the spatial distribution of vortex structures becomes increasingly compact with the enlargement of the rounded corner radius.

ENSO‐Modulated Variability in Winter Shelf Circulation of the Northern South China Sea

AbstractWe combined long‐term observational data from 1999 to 2021 with numerical simulations to study the seasonal changes in the currents along the continental shelf in the Northern South China Sea (NSCS) during winter and how these changes relate to the El Niño‐Southern Oscillation (ENSO). Our results indicate that during El Niño events, the eastward movement of the warm pool in the tropical Pacific Ocean not only leads to colder sea surface temperatures and a stronger Kuroshio current intrusion but also significantly affects the formation of a high‐pressure system in the subtropics over the NSCS. In El Niño years, an anomalous anticyclonic wind stress curl in the South China Sea weakens the northeast winter monsoon, which in turn weakens the cyclonic shelf circulation. The first principal component from the multivariate empirical orthogonal function decomposition, which accounts for 49.02% of the total variance, shows a significant correlation of 0.64 with the Niño 3.4 index, indicating the circulation's sensitivity to tropical climate changes. Our analysis of the winter shelf circulation, based on the along‐isobath depth‐integrated vorticity equation, reveals that the exchange of water across the isobaths over the shelf is mainly controlled by the nonlinear advection of relative vorticity, with wind stress curl and bottom stress curl playing a less significant role in regulating the structure of these exchanges. The combined effect of baroclinic forces and topography likely governs the dynamics over the slope.

On space–time diversity in shock train self-excited oscillation mode during wide-range evolution in a scramjet isolator

The shock train self-excited oscillation can induce combustor instabilities and reduce engine margin. In a dual-mode scramjet, the shock train undergoes a complete evolution process, exhibiting structural changes closely tied to this inherent unsteadiness. This study aims to elucidate the space–time diversity in shock train self-excited oscillation mode and the underlying mechanisms during wide-range evolution. The experimental investigations were conducted at Ma = 1.95, capturing the complete evolution of the shock train. The results indicate the evolution can be categorized into three regimes based on structural characteristics. In regime I, the shock region gradually forms, followed by the occurrence of the mixing region in regime II. Regime III corresponds to inlet unstart. In regime II, isolator outlet pressure fluctuations exhibit higher frequency and lower amplitude compared to regime I, while the shock motion demonstrates lower frequency and higher amplitude. The shock train behaves in a large-scale, low-frequency (1.53 times the duct height, 10 Hz) unsteady motion in regime II, posing a potential threat to engine operation. Coherence and phase analysis reveal the disturbance source originates downstream. Proper orthogonal decomposition modal analysis shows two oscillation modes: low-frequency components correspond to shock motion, and high-frequency components correspond to pressure fluctuations across the entire pseudoshock. The propagating of downstream disturbance differs between the two regimes. In regime I, the shock train exhibits rigid-body motion synchronously. In regime II, the relative motion between each shock wave and the cumulative effect of pressure disturbance lead to frequency decay upstream, amplifying the shock train motion.

A reduced-dimension method of Crank-Nicolson finite element solution coefficient vectors for the unsteady Burgers equation

This paper primarily focuses on the dimensionality reduction of finite element (FE) solution coefficient vectors for the unsteady Burgers equation, solved using the Crank-Nicolson FE (CNFE) method. The proper orthogonal decomposition (POD) basis is constructed from the snapshot matrix, which is formed using the first L solutions, where L is significantly smaller than the total number of time steps N of the CNFE method. By reconstructing the matrix form of the CNFE method, a reduced-dimension Crank-Nicolson finite element (RDCNFE) method is proposed and stability analysis and error estimates are discussed. Numerical tests are implemented to verify the theoretical results and demonstrate the high efficiency of the RDCNFE method.

Highly separated transitional shock-wave/boundary-layer interactions: A spatial modal study

At cruise altitudes, the Reynolds number may become sufficiently low to allow a laminar boundary layer to persist on the suction side of a transonic fan blade up to the shock-wave/boundary-layer interaction (SBLI). In such a transitional SBLI with sufficiently large shock-induced separation, a shock oscillation mechanism occurs, with the source at the upstream growth (until a critical length) and natural suppression (through shear layer instabilities) of the separation bubble. The oscillation cycle is characterized by a temporarily vanishing upstream laminar part of the separation bubble. The suppression of this laminar part is accompanied by downstream advection of turbulence and subsequent entrainment into the bulk separation bubble, affecting the reflected shock movement. In order to study the spatial behavior of the mechanism, particle image velocimetry of a highly separated transitional oblique SBLI at Mach 2.3 is conducted in the high speed aerodynamics laboratory of Delft University of Technology. Statistical quantities, including root mean square velocity fluctuations, phase averages, and spatial modes from proper orthogonal decomposition, are investigated. The entrainment strength varies depending on the phase of the oscillation, and the turbulent shear layer is not fully developed. The main growth and shrinking mode of the separation bubble were extracted, which affects the slip line region size and shock position. Secondary modes that affect the shear layer undulation and upstream effects were also extracted. The study provides quantitative analyses of an important shock oscillation type, with the focus on capturing the separation bubble size variation and upstream effects.