Iterative Equation Research Articles

The classical iterative HHL-based hemodynamic simulation quantum linear equation algorithm for abdominal aortic aneurysm

The classical iterative HHL-based hemodynamic simulation quantum linear equation algorithm for abdominal aortic aneurysm

Phase transition analysis of the Potts-SOS model with spin set {−1,0,+1} on the Cayley tree

Abstract Recently, the Potts-SOS model on Cayley trees was studied using the Kolmogorov consistency condition to investigate the existence and properties of Gibbs measures. The author previously examined the thermodynamic properties of the one-dimensional Potts-SOS model on the positive natural number lattice N , demonstrating the absence of phase transitions [Akin H, Phys. Scr. 99 (2024), 055 231]. In this study, we apply the cavity method to explore the non-uniqueness of Gibbs measures for the Potts-SOS model with spins {–1, 0, + 1} on a second-order semi-infinite Cayley tree. We address the phase transition problem by analyzing the model’s iterative equation system and perform stability analysis at fixed points to determine regions where phase transitions occur.

One-shot omnidirectional pressure integration through matrix inversion

Abstract In this work, we present a method to perform 2D and 3D omnidirectional pressure integration from velocity measurements with a single-iteration matrix inversion approach. This work builds upon our previous work, where the rotating parallel ray approach was extended to the limit of infinite rays by taking continuous projection integrals of the ray paths and recasting the problem as an iterative matrix inversion problem. This iterative matrix equation is now ``fast-forwarded'' to the ``infinity'' iteration, leading to a different matrix equation that can be solved in a single {\color{blue}step}, thereby presenting the same computational complexity as the Poisson equation. We observe computational speedups of $\sim10^6$ when compared to brute-force omnidirectional integration methods, enabling the treatment of grids of $\sim 10^9$ points and potentially even larger in a desktop setup at the time of publication. Further examination of the boundary conditions of our one-shot method shows that omnidirectional pressure integration implements {\color{blue} a boundary condition where the boundary points are treated as interior points to the extent that information is available}. Finally, we show how the method can be extended from the regular grids typical of particle image velocimetry to the unstructured meshes characteristic of particle tracking velocimetry data.

Efficient magnetic forward modeling for strongly magnetized bodies: An iterative integral equation approach

Magnetic surveying encounters challenges in high-precision detection and inversion on strongly magnetized bodies. Nonuniform magnetization caused by self-demagnetization and mutual magnetization makes magnetic field calculation using analytical formulas impractical. The previous discrete numerical methods have suffered from computational inefficiency or low accuracy. To resolve this issue, we develop an algorithm to solve the magnetic field integral equation for calculating the magnetization state of high-susceptibility bodies, combining the improved spatial convolution forward approach with the adaptive relaxation iteration method. To address the excessive computational burden in the magnetization field between 3D voxel arrays of the discrete model, we construct the circular convolution kernel matrix directly when the subsurface space is divided with an equidistant grid, significantly reducing redundant computations. Then, the fast Fourier transform is used to accelerate the forward discrete circular convolution operation. In addition, we derive an iterative computation scheme that adaptively adjusts the relaxation factor based on magnetic field changes to correct the effective magnetization for algorithm convergence. From a pragmatic perspective, equating the cuboid to equal-volume sphere subdivisions is developed to enhance computational efficiency further by approximately 50% despite sacrificing slight accuracy. The sphere and spherical shell models validate the algorithm accuracy, efficiency, and convergence. Furthermore, we analyze the characteristics of the mutual magnetization effect among magnetic bodies under different magnetization conditions. Finally, the applicability of the algorithm is demonstrated with a realistic example. Our algorithm shows promise for precisely exploring highly magnetized minerals, underground pipelines, and unexploded ordnance.

Polynomial-like iterative equation on Riesz spaces

Polynomial-like iterative equation on Riesz spaces

One-Shot Omnidirectional Pressure Integration Through Matrix Inversion

In this work, we present a method to perform 2D and 3D omnidirectional pressure integration from velocity measurements with a single-iteration matrix inversion approach. This work builds upon our previous work, where the rotating parallel ray approach was extended to the limit of infinite rays by taking continuous projection integrals of the ray paths and recasting the problem as an iterative matrix inversion problem. This iterative matrix equation is now ``fast-forwarded'' to the ``infinity'' iteration, leading to a different matrix equation that can be solved in a single iteration, thereby presenting the same computational complexity as the Poisson equation. We observe computational speedups of $\sim10^6$ when compared to brute-force omnidirectional integration methods, enabling the treatment of grids of $\sim 10^9$ points and potentially even larger in a desktop setup at the time of publication. Further examination of the boundary conditions of our one-shot method shows that omnidirectional pressure integration implements a new type of boundary condition, which treats the boundary points as interior points to the extent that information is available.

Experimental study of a leakage location method based on plug flow mass transfer characteristics

Leaks in water supply pipes pose a threat to water security. In order to save labor investment in pipeline leak location and to find a method to obtain the leak location by calculating upstream and downstream related data, an experimental study was conducted to investigate the plug flow mass transfer characteristics in horizontal pipelines. The results show that the addition of carbon dioxide gas can better detect the occurrence of small area ruptures in the pipeline. The translational velocity equation for elongated bubbles has better accuracy for both the insoluble gas air and the slightly soluble gas carbon dioxide. The local volumetric mass transfer coefficient shows a better linear relationship with the residual gas volume. The iterative equation derived from such a law can effectively predict the distribution of carbon dioxide concentrations in the pipeline. The study shows that the iterative equation, combined with the carbon dioxide concentration at a point downstream and the bubble translation velocity, can make a good localization judgment of the pipeline rupture point.

3D Marine Controlled-Source Electromagnetic Numerical Simulation Considering Terrain and Static Effect

The marine controlled-source electromagnetic (CSEM) method is an important geophysical method for the exploration of marine hydrocarbon resources. In marine CSEM forward modeling the uplift terrain, such as submarine hills and seamounts, the static effects caused by polymetallic nodules and hydrothermal sulfide are ignored which can lead to the deviation of marine CSEM data. To improve the accuracy of data processing and interpretation, this study realizes efficient 3D numerical simulation considering submarine uplift terrain and static effect based on the finite difference (FD) algorithm in a fictitious wave domain. First, based on the correspondence principle between the fictitious wave domain and a real diffusive domain, we derive the FD electromagnetic field iteration equations of the fictitious wave domain, and apply the fictitious emission source and the boundary condition of complex frequency shifted-perfectly matched layer (CFS-PML). We use the inverse transformation method to convert the electromagnetic response to the diffusive domain. Then, we carry out simulations on typical 1D and 3D reservoir models to verify the correctness and effectiveness of the algorithm. Furthermore, we design an uplift terrain model and a static effect model and study the influence of parameters such as top width, bottom width, height and volume of uplift terrain on the CSEM field response characteristics through the forward modeling of multiple models and discuss the influence of parameters such as electrical conductivity, length, width, thickness, depth and volume of a shallow anomaly on the marine CSEM response. Finally, we analyze the characteristics and rules of electromagnetic field propagation of uplift terrain and static effect, which provides theoretical guidance for the design of marine CSEM exploration systems.

Central cone film cooling scheme of turbofan engine based on multi-fidelity simulation

To improve the computational efficiency of multi-fidelity simulation, an improved fully coupled method was proposed, which was applied to the coupled simulation of a 3-D model of an exhaust system and a 0-D model of a turbofan engine. Different from conventional approach, by “truncating” the engine model, the CFD calculation of the exhaust system can be decoupled from the solution of nonlinear equations in the engine model, and an outer iteration equation set was constructed to solve the multi-fidelity model. It was found that the improved fully coupled method has better computational efficiency than traditional methods while maintaining equivalent computational results. Then, the improved fully coupled method was applied to the design of the central cone film cooling system, the influence of cooling gas bleed mechanism, the hole open area ratio of the central cone and hole density, hole direction, hole shape on the infrared characteristics was investigated. The impact of central cone film cooling on the overall performance of the engine was quantitatively evaluated. Combined with various favorable factors, when maintaining the HP rotor speed unchanged, the integrated infrared radiation intensity of the central cone under supersonic cruise conditions is reduced by 93 % compared to the uncooled state, while the engine thrust increases by 0.3 % and fuel consumption rate increases by 0.18 %. The multi-fidelity simulation method proposed in this paper and the conclusions obtained are of great significance for the structural design of low-infrared exhaust systems.

Uniformly-like convex solutions for the polynomial-like iterative equation in Banach spaces

Uniformly-like convex solutions for the polynomial-like iterative equation in Banach spaces

Smooth solutions of a class of iterative functional equations

Smooth solutions of a class of iterative functional equations

Scattering properties of dual Bessel beams on chiral layered particle

The paper investigates the scattering analytical solution of arbitrary direction double zero-order Bessel beams (ZOBBs) incident on a non-uniform layered chiral sphere. Using the generalized Lorenz-Mie theory (GLMT), the electromagnetic field expansion expression of the double ZOBBs is obtained through the vector superposition of the spherical vector wave functions (SVWFs). Analytical solutions of the scattering on chiral layered particles by dual Bessel beams are derived based on the boundary condition. The iterative coefficient equations are constructed to obtain the expansion coefficients of the SVWFs in each region of the multi-layer chiral sphere. The variation of radar cross section (RCS) with angular distribution is numerically simulated, and the scattering results of layered chiral spheres degenerated into isotropic layered spheres are compared with the literature, which verifies the correctness of the theory and program. The effects of incident angle, polarization angle, cone angle and chiral parameters on the scattering characteristics of far region and internal field of the layered chiral particle are analyzed in detail. The theory and results can offer significant assistance in studying the optical properties of non-uniform chiral particles and complex biological cells irradiated by multiple beams and may also provide important practical value in the micromanipulation of multi-layered biological structures.

Existence and uniqueness of continuous solutions for iterative functional differential equations in Banach algebras

Existence and uniqueness of continuous solutions for iterative functional differential equations in Banach algebras

Real-time improvement method of turbo-shaft engine component-level model based on dimensional reduction of residual iterative equation

This paper explores a method to improve the real-time computation performance of turbo-shaft engine component-level model by reducing the dimensionality of residual equations set. For turbo-shaft engine, nozzle typically operates in a subcritical working state, assuming that the gas total temperature and pressure parameters of power turbine inlet are the same, and by assuming the gas flow balance between gas generator outlet and nozzle inlet, the iterative solution of the pressure drop of nozzle can be obtained. Then the pressure ratio of power turbine is naturally obtained, thereby achieving the objective of dimensionality reduction of the residual equations set. The component-level model’s steady-state and transient performance were simulated on a desktop computer with a clock frequency of 2.6 GHz. Simulation results show that, with a 2% margin of error, the steady state computation time for the component-level model was reduced by 54.8%, while the transient performance computation time was reduced by 18.6%. This represents a noticeable improvement in real-time performance. Simulation results on the P2020 embedded processor platform with an 800MHz clock frequency indicate a 22.3% reduction in single-step maximum transient performance computation time of the reduced dimensional model, compared to the original model. This demonstrates the effectiveness of dimensionality reduction of the residual equations set in enhancing the model real-time computation performance.

Dynamic-mode-decomposition-based gradient prediction for adjoint-based aerodynamic shape optimization

Accurate and efficient gradient computation is the key to aerodynamic shape optimization. In this paper, dynamic mode decomposition (DMD) is employed to analyze the dynamic characteristics of the early pseudo-time marching of adjoint equations and to predict the gradient. Besides the first-order zero-frequency mode, other zero-frequency modes also contribute to the pseudo iterations of the adjoint equations in the early iterations. Hence, different from existing methods, all zero-frequency modes are retained to reconstruct adjoint fields for gradient prediction. Moreover, to further improve the modeling accuracy, an improved DMD (IDMD) is proposed by omitting the initial snapshots in early iterations. The effect of pseudo-time step on modeling accuracy is also studied. By solving the adjoint equations of the transonic and subsonic flows, the accuracy of the proposed method is verified. Results indicate that the proposed method still works despite the conventional solution process diverges. Through aerodynamic shape optimization examples of transonic flow over an airfoil, the number of adjoint pseudo-time steps is remarkably reduced by 83%, which indicates the proposed IDMD-based gradient prediction method has great potential for improving the efficiency of aerodynamic shape optimization.

Fractional order iterative boundary value problem

In the present paper, we establish the existence and uniqueness solution for an iterative differential equation involving caputo fractional derivative of order $1\alpha2$. The existence results is proved by using Schauder fixed point. To prove the extremal solution, we demonstrate some fractional inequalities. As application, we conclude this paper by giving an illustrative example to demonstrate the applicability of the obtained result.

Application of spatial filtering high order FDTD method in the multi-physics field coupled equations

The traditional finite-difference time-domain method (FDTD(2, 2)) is widely employed to solve the coupled equations of quantum-electromagnetic multi-physics. However, the time step of the traditional explicit FDTD(2, 2) method is limited by the Courant-Friedrichs-Lewy (CFL) condition. Besides, the FDTD(2, 2) method only achieves the second-order accuracy, which can result in significant error accumulation. To overcome this problem, we propose the SF-HO-FDTD(2, q) method with scalable time stability conditions for solving the multi-physics field coupled equations (q denotes q-th order spatial difference scheme). This approach integrates spatial filtering (SF) and higher-order FDTD (HO-FDTD) methods. Moreover, the proposed method requires no further derivation of the iterative equations of the traditional HO-FDTD(2, q) method. It only needs to introduce the SF operation in each numerical iteration process to filter out the spatial high-frequency components arising from the use of time steps that do not satisfy the CFL stability condition. This condition is applied to ensure the stability of the numerical method. Therefore, the proposed method has a high compatibility with the traditional HO-FDTD(2, q) method. Then, the numerical dispersion analysis of the SF-HO-FDTD(2, q) method is theoretically studied. Finally, numerical examples are carried out to confirm the accuracy and efficiency of the strategy suggested in this study.

Dynamic phasor measurement algorithm based on high-precision time synchronization

Ensuring the swift and precise tracking of power system signal parameters, especially the frequency, is imperative for the secure and stable operation of power grids. In instances of faults within the distribution network, abrupt changes in frequency may occur, presenting a challenge for existing algorithms that struggle to effectively track such signal variations. Addressing the need for enhanced performance in the face of frequency mutations, this paper introduces an innovative approach—the Covariance Reconstruction Extended Kalman Filter (CREKF) algorithm. Initially, the dynamic signal model of electric power is meticulously analyzed, establishing a dynamic signal relationship based on high-precision time source sampling tailored to the signal model’s characteristics. Subsequently, the filter gain, covariance matrix, and variance iteration equation are determined based on the signal relationship among three sampling points. In a final step, recognizing the impact of the covariance matrix on algorithmic tracking ability, the paper proposes a covariance matrix reset mechanism utilizing hysteresis induced by output errors. Through extensive verification with simulated signals, the results conclusively demonstrate that the CREKF algorithm exhibits superior measurement accuracy and accelerated tracking speed when confronted with mutating signals.

STODOLA-VIANELLO ITERATION METHOD FOR FREE TORSIONAL VIBRATION ANALYSIS OF MONOSYMMETRIC BOX-BEAM BRIDGES

Box-beam bridges are used in large spans and wider decks due to their high strength and greater torsional and flexural stiffness. They are usually prone to vibration due to moving vehicular traffic. Their eigenfrequency analysis is a crucial aspect of their design in order to ensure that the natural frequency is not close to the excitation frequency to avert resonance failures. The free torsional vibration equation is a fourth order partial differential equation (PDE) with variable parameters. For prismatic cross-sections, homogeneous and isotropic materials the governing PDE have constant parameters. This paper explores the Stodola-Vianello iteration method (SVIM) for solving the PDE for isotropic, homogeneous prismatic box-beams. Harmonic response is assumed, decoupling the PDE to two equations, one in terms of time and the second an ordinary differential equation (ODE) in terms of space coordinates. The Stodola-Vianello method is used by the method of four successive integrations to express the ODE as an algebraic iteration problem with four constants of integration. The four boundary conditions are used to solve for the four constants of integration, thus making the problem determinate. Application of the boundary conditions results in the full determination of the iteration equations. For simply supported boundaries studied in the work, a trigonometric buckling shape function that satisfies all the boundary conditions is employed in the SVIM formula to obtain the next bucking modal shape function. The requirement for convergence is then used to establish the characteristic buckling equation from which the eigenvalues are obtained. The solution to the characteristic buckling equation gave the exact mathematical expression for the natural torsional frequency at the nth vibration mode. The frequencies are determined for the first eight vibration modes and compared with previously obtained values. It was found that the natural frequencies are identical with previously found values of the exact natural frequencies. The natural frequency obtained is exact because it satisfies the PDE and the boundary conditions at all points in the domain.

Exact results for gene-expression models with general waiting-time distributions.

Complex molecular details of transcriptional regulation can be coarse-grained by assuming that reaction waiting times for promoter-state transitions, the mRNA synthesis, and the mRNA degradation follow general distributions. However, how such a generalized two-state model is analytically solved is a long-standing issue. Here we first present analytical formulas of burst-size distributions for this model. Then, we derive an iterative equation for the mRNA moment-generating function, by which mRNA raw and binomial moments of any order can be conveniently calculated. The analytical results obtained in the special cases of phase-type waiting-time distributions not only provide insights into the mechanisms of complex transcriptional regulations but also bring conveniences for experimental data-based statistical inferences.