Orthogonal Decomposition Method Research Articles

Error analysis of a reduced order method for the Allen-Cahn equation⁎

In this paper we carry out an error analysis for a reduced order method for the Allen-Cahn equation. First, an ensemble of snapshots is formed from the numerical solutions at some time instances of the full order model, which is a time-space discretisation of the Allen-Cahn equation. The reduced order model is essentially a new spatial discretisation method by using low dimensional approximations to the original approximation space. The low dimensional approximation space is generated from the ensemble of snapshots by applying a proper orthogonal decomposition method. To determine the error between the exact solution and the solution of the reduced order model. We consider a time-space discretisation for which an error estimate of the full model solution is available. Specifically, the full discretisation is based on a stabilized auxiliary variable approach for the time stepping and a spectral Galerkin method for the spatial discretisation. The advantages of this full discretisation are its unconditional stability, the availability of error estimates and its ease of implementation. An estimate of the errors in the H1 seminorm is rigorously derived for both the full order model and the reduced order model, which is then verified by some numerical examples.

Proper orthogonal decomposition method of constructing a reduced-order model for solving partial differential equations with parametrized initial values

This paper proposes a novel proper orthogonal decomposition method for constructing a reduced-order model. This model effectively computes solutions for various initial conditions in time-dependent partial differential equations. The mode obtained from the proposed proper orthogonal decomposition captures both the fluctuating component and the mean-field of the time-dependent solution. In this mode, the eigenvalues representing the mean-field, are significantly larger than those representing the fluctuating component. Consequently, determining the number of modes required for representing a solution cannot rely solely on the cumulative contribution rate. Therefore, the proposed method gives a scalar weight to the mean-field and controls the magnitude of the mean-field energy. We propose a method that considers the magnitude of the scaler weight alongside the cumulative contribution ratio for determining the number of modes. This method was evaluated using the time-dependent Burgers equation with parametric initial values. Our proposed method selects the appropriate mode to represent the given dataset, unlike the conventional method. A reduced-order model based on the proposed method effectively computes the time evolutions of the solutions using datasets for several parameters by imposing a condition that the cumulative contribution ratio exceeds 98%.

Experimental investigation on the effects of transverse injection distribution scheme on dual flame dynamics subjected to flow disturbances

Combustion instability has become a major problem in low-emission annular gas turbine combustors. It is of practical importance to investigate the effects of transverse nozzle distribution scheme on multi-flame dynamics subjected to external acoustic perturbations. In this study, the responses of heat release rate perturbations from dual propane/air laminar premixed flames to axial acoustic perturbations have been experimentally measured. Specifically, the velocity disturbance upstream the flame base has been measured by two-microphone method and the flame heat release rate perturbation is obtained by the photomultiplier tube with CH* filter lens. The dynamic evolution of dual flames is captured by the high-speed camera and then analyzed with the proper orthogonal decomposition (POD) method. Results indicate that there is a critical distribution distance, where the peak gain of the dual flame transfer function is the largest. And the direct injection dual flame transfer function can behave as the low-pass filter. Furthermore, the dual flame transfer function behaves nonlinearly with the increase of the injection angle. The dual flame transfer function for large injection angle can behave as the hybrid mode of the low-pass filter and the band-pass filter. The POD analysis reveals that the coupling effect of gas thermal expansion and stretch effect would contribute to the interaction between dual flame front dynamics. The transverse injection distribution scheme (dual flame distance and injection angle) has a significant effect on the dual flame front dynamics, which in turn affects the flame heat release rate disturbance. The bulk mean flow velocity can enhance the interaction effect between dual flame front dynamics.

Open-loop and closed-loop local and remote stochastic nonzero-sum game with inconsistent information structure

In this paper, the open-loop and closed-loop local and remote stochastic nonzero-sum game (LRSNG) problem is investigated. Different from previous works, the stochastic nonzero-sum game problem under consideration is essentially a special class of two-person nonzero-sum game problem, in which the information sets accessed by the two players are inconsistent. More specifically, both the local player and the remote player are involved in the system dynamics, and the information sets obtained by the two players are different, and each player is designed to minimise its own cost function. For the considered LRSNG problem, both the open-loop and closed-loop Nash equilibrium are derived. The contributions of this paper are given as follows. Firstly, the open-loop optimal Nash equilibrium is derived, which is determined in terms of the solution to the forward and backward stochastic difference equations (FBSDEs). Furthermore, by using the orthogonal decomposition method and the completing square method, the feedback representation of the optimal Nash equilibrium is derived for the first time. Finally, the effectiveness of our results is verified by a numerical example.

Investigation of the unsteady pressure fluctuation mechanism in a regenerative flow pump based on proper orthogonal decomposition

The regenerative flow pump (RFP) derives its name from the circular flow pattern akin to a vortex. However, the presence of numerous vortices results in highly turbulent flow and pressure fluctuation. This study delves into the characteristics of the internal unsteady flow in two distinct RFP models featuring different blade shapes. Numerical simulations are employed to obtain the pressure field, which is subsequently scrutinized using the proper orthogonal decomposition (POD) method. The results show that the average pressure in the peripheral direction undergoes minimal variation during the flow developing stage, experiences a sharp increase in the fully developed stage, and finally exhibits substantial changes in the stripper. The transient pressure in the time domain fluctuates periodically, and the dominant blade passing frequency in the frequency domain demonstrates a similar trend along the circumferential direction as the average pressure. Moreover, the fluctuating intensity of pressure diminishes along the impeller's rotating direction but intensifies significantly in the stripper. Comparative analysis indicates that the exchange intensity is influenced by the flow at different developmental stages, and the exchange flow conditions could reflect the fluctuating intensity. Furthermore, the study reveals that the frequency amplitude of the time coefficient gradually decreases as the mode order increases. The first and second modes exhibit a gradually changing trend associated with pressure increase patterns, whereas the third and fourth modes highlight the emergence of localized modulation phenomena linked to exchange flow. Thus, the POD method offers a unique perspective for comprehending the flow mechanisms within RFPs.

Experimental and numerical study on buffeting force characteristics of the π -shaped bridge deck

In this study, a π-shaped main beam with typical geometric characteristic parameters was selected for conducting wind tunnel tests, and the characteristics of the buffeting force were measured. Based on the measured results, numerical expansion research was conducted using the narrowband synthetic random flow generation (NSRFG) turbulent inlet method, and a grid strategy was provided. By changing the geometric characteristic parameters of the π-shaped girder, a comparative study was conducted using proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) methods, revealing the influence of cross-sectional geometric characteristic parameters on the buffeting force characteristics and analyzing their mechanism of action. The results indicate that the inlet wind parameters of the NSRFG need to be adapted to the grid size. The grid filter size at the front end of the model should be smaller than 0.193 of the along-wind turbulence integral scale, which can then be used to solve for 80% of the turbulent kinetic energy. The smaller the aspect ratio is, the larger the buffeting force spectrum is, and the smaller the opening ratio is, the smaller the buffeting force spectrum is. The opening ratio strongly influences the buffeting lift spectrum, and the aspect ratio strongly influences the buffeting drag spectrum. The POD decomposition indicates that the geometric characteristic parameters affect the shape, strength, position, and direction of vortices at the section opening. DMD decomposition indicates that geometric feature parameters affect the frequency and growth rate of dominant modes as well as the directionality and regularity of vortex distribution.

A physics merged deep neural network-based prediction method for jet turbulent mixing noise

Turbulent mixing noise is a vital component of jet noise, and its rapid, accurate prediction has always been persistently pursued. Recent advancement in machine learning has been applied to jet noise prediction. However, these applications are pure curve fitting and lack physical constraints. In this study, a physics-merged deep neural network (PMNN)-based prediction method was developed for turbulent mixing jet noise by merging the physics of the jet noise. This deep neural network (DNN)-based method employed recent advancements in jet turbulent mixing noise containing large- and fine-scale turbulence structures. Two simple rational functions for large- and fine-scale turbulent noise similarity spectra were proposed to replace the original complex similarity spectra functions and incorporated into the DNN-based prediction method. For comparison, we present two data-driven DNN-based prediction methods (DDNN). The first DDNN method used the sound pressure level (SPL) as the output of neural networks, directly establishing the nonlinear relationship between the input features and SPL. In the second DDNN method, the dominant modes of the jet noise spectra extracted using the proper orthogonal decomposition method were merged with DNN. These DNN-based methods were then trained using a set of experimental data over a wide range of jet operating conditions. Their performance was evaluated and compared. It was evident that all these DNN-based methods were capable of predicting turbulent mixing noise reasonably well. In contrast to the DDNN methods, the PMNN method could provide insights into the jet turbulent mixing noise components. It demonstrates that the turbulent mixing jet noise spectra at the mid polar angle is generated by the large-scale noise component at low-frequency range and by the fine-scale noise component at high-frequency range.

Domain Decomposition Strategy for Combining Nonlinear and Linear Reduced-Order Models

Increasingly ubiquitous reliance on expensive, high-fidelity numerical simulations has led to the emergence of reduced-order modeling as an effective method to predict high-dimensional, spatially discretized field outputs in an efficient manner. This study presents a method to construct parametric, nonintrusive reduced-order models (ROMs) by leveraging domain decomposition (DD) to enable using multiple dimension reduction (DR) techniques within different spatial subdomains of the field to construct a single predictive ROM. To enforce smoothness of solutions across subdomains, points at domain interfaces are reconstructed using the data-driven gappy proper orthogonal decomposition method. The method is assessed on three high-dimensional test cases with shocks, including one with two-dimensional turbulent flow around the RAE2822. Results show that fully nonlinear ROMs predict fields more accurately in the vicinity of discontinuous features compared to their linear counterparts, which are more accurate in regions that are far from these discontinuities. Furthermore, irrespective of the DR method, DD-based ROMs outperformed their global counterparts in all test cases. Finally, models that used a combination of DR methods simultaneously showed superior performance near shockwaves and a significant improvement in total error compared to existing approaches that employ a single DR method in all subdomains.

Adaptive Orthogonal Basis Function Detection Method for Unknown Magnetic Target Motion State

Traditional methods of orthogonal basis function decomposition have been extensively used to detect magnetic anomaly signals. However, the determination of the relative velocity between the detection platform and the magnetic target remains elusive in practical detection. And, the non-ideal uniform motion of the magnetic target further complicates the process and adversely impacts the detector’s performance. To address this challenge, this paper introduces an adaptive scale factor method based on orthogonal basis function decomposition. This new method can be used to adjust the relative velocity between the detection platform and the magnetic target and to refine the characteristic time in the basis functions. In this paper, a mathematical relationship between the scale factor and the relative velocity is established, which is subsequently fitted into a Gauss function curve. The optimal scale factor, denoted as β, is adaptively chosen from the fitting curve when the magnetic target moves at a non-ideal uniform velocity with an unknown motion state. The results of simulations indicate that the scale factor improves the signal-to-noise ratio of the magnetic anomaly signals in a non-ideal state. And, this method can improve the energy value of OBF decomposition by 17.7%. Simultaneously, this method ensures that the moment the magnetic target passes the CPA coincides with the energy peak of the orthogonal basis detection, which improves the accuracy by 54.1% compared with the traditional method. The effectiveness and precision of the proposed method are verified using simulations and practical experiments.

Dynamic modeling and parameter sensitivity analysis of AUV by using the POD method and the HB-AFT method

Autonomous Underwater Vehicle (AUV) parameterization is a geometric modification method, and can quickly obtain excellent structure with low vibration and noise performance. Blindness and inefficiency are common outcomes of the lack of an efficient parameter selection approach. But not enough research has been done in this area. In this paper, the finite element (FE) method is used to establish the propeller–shaft–shell double beam system with ball bearings, which is more similar to the real model. The basic theory of the proper orthogonal decomposition (POD) method is introduced in detail, and the original full order model (FOM) is reduced to 4 DOFs. It is found that the accuracy of the reduced order model (ROM) is high enough and the calculation time is greatly reduced. The semi-numerical semi-analytical expression of the ROM system is solved by the harmonic balance and alternating frequency/time domain (HB-AFT) method, whose results are in good agreement with those by the Runge–Kutta method. The sensitivities of the shell responses to the spring stiffness are discussed. The AUV setup is designed to analysis of numerical and experimental results. These results in this paper can provide an advanced theoretical basis for the structural parameter design of the AUV.

Automatic Proper Orthogonal Block Decomposition method for network dynamical systems with multiple timescales

In this work, we introduce a novel reduced order model technique, based on the Proper Orthogonal Decomposition method, for dynamical systems with multiple timescales. The main ideas are to retain the structure of the original model, which is lost in the original POD procedure, while producing a competitive reduction in the number of equations and computational time, and to determine the best structure for the reduced system automatically, via a data-driven analysis of the original model data. For these novel techniques, we present some numerical tests for various behaviors of three different neural network models with multiple timescales, which support the use of these new methods.

POD method for solving elastodynamics problems of functionally gradient materials based on radial integration boundary element method

Functionally graded materials (FGMs) are widely utilized due to their excellent mechanical properties, prompting a need to explore the dynamic response of FGMs. To accurately and efficiently analyze elastodynamics problems, the radial integration boundary element method (RIBEM) is coupled with the proper orthogonal decomposition (POD) method in this paper. The POD method can transform a high-dimensional model into a low-dimensional system with high precision. Using the basic solution (Kelvin solution), the governing partial differential equation for the linear elastic dynamics problem of FGMs is transformed into a boundary-domain integral equation. The radial integration method is employed to convert the domain integral arising from material inhomogeneity and inertia terms into an equivalent boundary integral. This establishes a solution for the elastic dynamics problem of FGMs without the need for internal grid RIBEM. The Newmark time integration scheme is applied to solve the second-order ordinary differential equations. The displacement field obtained by RIBEM is utilized to construct the snapshots matrix, from which the POD reduced-order model is established. Numerical examples validate that the POD method significantly reduces the model's order, enhancing computational efficiency while also demonstrating the high accuracy of the POD method.

Influence of turbulent coherent structures on the performance and wake of a wind turbine

Wind energy in the atmospheric boundary layer serves as the primary source for energy absorption and structural load on wind turbines. However, the impact of turbulent coherent structures on the aerodynamic performance and wake characteristics of wind turbines has not been comprehensively evaluated. In this study, the proper orthogonal decomposition (POD) method is employed to assess the influence of turbulent coherent structures of varying scales on the aerodynamic performance and wake characteristics of wind turbines in the neutral atmospheric boundary layer. The results show that turbulent coherent structures are the main factor that determines the wind velocity fluctuation, aerodynamic performance and wake characteristics of wind turbine in the atmospheric boundary layer. When considering the 13th or lower order POD mode, the wind velocity fluctuation increases with the increase of energy content (more POD modes) of the turbulent coherent structures. When considering the first 19 POD modes, the dynamic loads and power of wind turbine fluctuate with high frequencies, the thrust fluctuates in an amplitude range between 2.4 % and 13.9 % around the mean value, and the power fluctuates from 4.5 % to 28.6 % of the mean value. When considering the first 40 POD modes, the average power generation of the wind turbine increases by 26 % compared to the case with no turbulent structures considered. The study of turbine wake shows that turbulent coherent structures can expand the wind turbine wake approximately to a width of 2.5D and a height of 3D (D is the diameter of the wind turbine), offset the wake approximately to 2D, and move forward the position of the wake vortex beginning to dissipation approximately to 7D behind the wind turbine. In addition, turbulent coherent structures can accelerate the wake velocity recovery by increasing the momentum exchange between the atmospheric boundary layer and wind turbine wake.

Wake suppression of a cylinder immersed in turbulence using rotating rods

In this study, we conducted three-dimensional direct numerical simulations to investigate the control performance of the free-stream turbulence past a circular cylinder with two small rotating rods at Re = 3900. In the case A and case B, two control rods with different rotating ratio α are positioned symmetrically at the circumferential angle θ=±45° and θ=±120° from the forward stagnation point of the main cylinder, respectively. Here, α is the ratio of the tangential velocity of the control rod surface ur to the inflow velocity U∞. Compared to the bare cylinder immersed in turbulence, both installation positions of the rotating rods at α = 2 result in a significant suppression on the wake flow. Specifically, the time-averaged drag coefficient Cd¯ was reduced by 25%, and the root mean square value of the lift coefficient measured to reduce by up to 30%. Statistical analysis is then performed, in terms of the Reynolds stresses, mean field, and the turbulent wake visualization to show variations in the flow dynamics. As expected, the front-mounted control rods inject kinetic energy into the boundary layer, effectively suppressing the turbulence fluctuations on both sides of the main cylinder. The work done by viscous forces around the rotating control rods and the suppression of turbulent fluctuations contribute to the pressure recovery observed in the rear-mounted control rod case. Proper orthogonal decomposition method is further employed to analyze the key features of the controlled wake of two cases with α = 2.

Numerical investigation on the cavitating wake flow around a cylinder based on proper orthogonal decomposition

The non-cavitating and cavitating wake flow of a circular cylinder, which contains multiscale vortices, is numerically investigated by Large Eddy Simulation combined with the Schnerr–Sauer cavitation model in this paper. In order to investigate the spatiotemporal evolution of cavitation vortex structures, the Proper Orthogonal Decomposition (POD) method is employed to perform spatiotemporal decomposition on the cylinder wake flow field obtained by numerical simulation. The results reveal that the low-order Proper Orthogonal Decomposition modes correspond to large-scale flow structures with relatively high energy and predominantly single frequencies in both non-cavitating and cavitating conditions. The presence of cavitation bubbles in the flow field leads to a more pronounced deformation of the vortex structures in the low-order modes compared to the non-cavitating case. The dissipation of pressure energy in the cylinder non-cavitating wake occurs faster than the kinetic energy. While in the cavitating wake, the kinetic energy dissipates more rapidly than the pressure energy.

Complex Modal Characteristic Analysis of a Tensegrity Robotic Fish's Body Waves.

A bionic robotic fish based on compliant structure can excite the natural modes of vibration, thereby mimicking the body waves of real fish to generate thrust and realize undulate propulsion. The fish body wave is a result of the fish body's mechanical characteristics interacting with the surrounding fluid. Thoroughly analyzing the complex modal characteristics in such robotic fish contributes to a better understanding of the locomotion behavior, consequently enhancing the swimming performance. Therefore, the complex orthogonal decomposition (COD) method is used in this article. The traveling index is used to quantitatively describe the difference between the real and imaginary modes of the fish body wave. It is defined as the reciprocal of the condition number between the real and imaginary components. After introducing the BCF (body and/or caudal fin) the fish's body wave curves and the COD method, the structural design and parameter configuration of the tensegrity robotic fish are introduced. The complex modal characteristics of the tensegrity robotic fish and real fish are analyzed. The results show that their traveling indexes are close, with two similar complex mode shapes. Subsequently, the relationship between the traveling index and swimming performance is expressed using indicators reflecting linear correlation (correlation coefficient (Rc) and p value). Based on this correlation, a preliminary optimization strategy for the traveling index is proposed, with the potential to improve the swimming performance of the robotic fish.

An iterative proper orthogonal decomposition method for a parabolic optimal control problem

An iterative proper orthogonal decomposition method for a parabolic optimal control problem

Investigation of the free vibrations of bi-directional functionally graded material beams using differential quadrature method

In this study, we investigated the natural dynamic characteristics of bi-directional (2D) functionally graded material (FGM) beams using differential quadrature method (DQM). The higher order shear deformation theory was used to reconstruct the displacement and strain fields at any point on the beam. The resulting equilibrium equation of elasticity was expressed by stress is equivalently integrated along the height direction of a rectangular beam, which helps derive the ordinary differential characteristic equations of the free vibration problem of 2D FGM beams, including the natural frequency and the triple-coupled mode functions. Then, using the DQM, the eigenvalue problems of the derived differential equation for free vibration of 2D FGM beams were transformed into a set of algebraic equations representing the eigenvalue problems. The obtained algebraic equations were solved using the orthogonal triangular decomposition method (QR). First, several orders of natural frequencies of the free vibrations of 2D FGM beams were calculated sequentially, and then, the triple-coupled mode functions were obtained together. Numerical examples were used to illustrate the effects of the thickness ratio, grading indexes, and the supported edges on the natural frequency in detail. Numerical results validated the feasibility and accuracy of the developed DQM for solving the problem of free vibrations in 2D FGM beams.

Unsteady Flow Structure of Corner Separation in a Highly Loaded Compressor Cascade

Abstract Corner separation is an inherently unsteady flow feature in an axial compressor cascade, and it significantly affects the aerodynamic performance of compressors. The flow field of a highly loaded compressor cascade at the Mach number of 0.59 under the moderate separation condition is simulated based on delayed detached eddy simulation. Comparisons of averaged flow field and transient flow field show that the three-dimensional corner separation flow is highly unsteady and composed of fine-scale vortex structures. The classical recognition of corner separation structures is a consequence of time-averaging. To better understand the contribution of unsteady structures to the averaged flow structures, the evolutions of flow fields in time series and the power spectrums are analyzed. A dominant periodic flow fluctuation is caused by the development of separating vortices with a characteristic frequency around 3500 Hz or at a Strouhal number of 0.75. Further, energy scales and spatiotemporal features of these dominant unsteady behaviors are analyzed using proper orthogonal decomposition and dynamic mode decomposition methods. Results show that the low-frequency behaviors mainly caused by the passage vortex at lower-span regions govern large-scale changes of separation flow in size and intensity and act with certain intermittency. The vortex developing mode around 3500 Hz prevails at higher regions affected by the concentrated shedding vortex. As the separating vortices dissipate approaching the midspan, the effect of the vortex developing mode on axial velocity fluctuation is reduced, although it dominates the pressure fluctuation with good stability in the whole passage.

A reduced-order Jacobi spectral collocation method for solving the space-fractional FitzHugh–Nagumo models with application in myocardium

PurposeThe space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.Design/methodology/approachThe fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.FindingsA fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.Originality/valueThe spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.