Iterative Method Research Articles

A nonlinear interval finite element method for elastic–plastic problems with spatially uncertain parameters

This paper proposes a nonlinear interval finite element method for elastic–plastic analysis of structures with spatially uncertain parameters. The spatially uncertain parameters are described by the interval field, and the variation bounds of the elastic–plastic structural responses can be calculated effectively. Quantified by the interval field, the spatially uncertain parameters are represented by the interval Karhunen–Loève (K-L) expansion, based on which the nonlinear interval finite element equilibrium equation is formulated. An interval iterative method is then presented to solve the equilibrium equation and obtain an outer solution of the variation bounds of structural responses such as displacement. In this method, the Newton-Raphson iterative method is used to transform the nonlinear problem into a linear one, and then the interval iterative method is introduced to solve the interval linear equations. Three numerical examples are employed to illustrate the feasibility and accuracy of the proposed method.

Nonlinear vibrations of temperature-dependent FGM cylindrical panels subjected to instantaneous surface heating

An analysis is performed in this research to deal with the large amplitude vibrations caused by rapid surface heating in an FGM cylindrical panel. It is assumed that the properties of the panel are distributed through the thickness in terms of volume fraction of constituents. Following the first-order shear deformation theory, von Karman type of kinematics, and Donnell kinematic assumptions, definition of kinetic and strain energies of the shell are established. These obtained equations are discretized via the method of Ritz, where the shape functions are estimated by the Chebyshev polynomials. Temperature distribution within the body is also obtained using the one-dimensional heat conduction equation. This equation is discretized using the finite different method and solved iteratively following the Picard and Crank-Nicolson methods. Thermally induced force and moment resultants are evaluated and inserted into the matrix representation of equations of motion. The temporal evolution of displacement is obtained using the iterative Picard method and Newmark time marching method. Results of this study are compared with the available data in the open literature and after that novel results are given to explore the effects of geometrical, mechanical, and thermal parameters. It is highlighted that power law index, shallowness, thickness, edge supports, temperature dependency, geometrical nonlinearity and thermal boundary conditions are all important factors on the response of the structure under thermal shock. It is highlighted that thermally induced vibrations indeed exists especially in thin class of shells. Besides, thermally induced deflections may be controlled through using a proper power law index.

Dynamical and numerical analysis of the hepatitis B virus treatment model through fractal–fractional derivative

Infection with the hepatitis B virus (HBV) is a global health problem and may be controlled via appropriate treatment. We use fractional models to understand infectious diseases because fractional models help us understand treatments' effects on hepatitis B better than integer‐order models. In this article, we introduce a new mathematical model for HBV based on the fractal–fractional derivative with a generalized Mittag–Leffler kernel. Firstly, we discuss the fundamental properties, like the equilibria of the model and the primary reproduction number. Then, we investigate the existence of a unique solution to our model. We use an iterative method to solve the proposed model. Further, we discuss the stability of the model through stability theory. Finally, we offer some graphical illustrations for various values of the parameters.

Two Variants of Robust Two-Step Modulus-Based Matrix Splitting Iteration Methods for Mixed-Cell-Height Circuit Legalization Problem

Two Variants of Robust Two-Step Modulus-Based Matrix Splitting Iteration Methods for Mixed-Cell-Height Circuit Legalization Problem

Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative

Abstract In this work, we consider a class of fuzzy fractional delay integro-differential equations with the generalized Caputo-type Atangana-Baleanu (ABC) fractional derivative. By using the monotone iterative method, we not only obtain the existence and uniqueness of the solution for the given problem with the initial condition but also give the monotone iteration sequence converging to the unique solution of the problem. Furthermore, we also give the continuous dependence of the unique solution on initial value. Finally, an example is presented to illustrate the main results obtained. The results presented in this study are new and open a new avenue of research for fuzzy fractional delay integro-differential equations with the generalized ABC fractional derivative.

Machine Learning Augmentation of the Failure Assessment Diagram Methodology for Enhanced Tubular Structures Integrity Evaluation

Failure assessment diagrams (FADs) are essential engineering tools for evaluating the structural integrity of components. However, their widespread application can be limited by complexity and computational expense. This study presents a novel machine learning-based approach to streamline FAD analysis, offering accuracy and efficiency while overcoming these limitations. The approach integrates numerical contour integral-based FADs with artificial neural networks (ANNs). To ensure reliable material modeling for the Finite Element Analysis (FEA) used to generate J-integral based FADs that train the ANNs, careful experimental and numerical procedures were employed. This involved uniaxial tensile tests, an iterative method for obtaining precise true stress–strain curves, and a Ramberg–Osgood material model for accurate material behavior representation. The ANNs themselves not only analyze large datasets to generate precise FAD envelopes but also predict limit loads and the Φ parameter, incorporating the effect of residual stress on the FAD methodology. To verify and test the proposed method, hypothetical fitness-for-service assessment cases were conducted, incorporating experimental residual stress measurements from split-ring tests on P110 and L80 pipes. These assessments were compared to both traditional FAD methods and computationally intensive FEA-based FADs. Results demonstrate a closer agreement with FEA-based calculations than traditional methods provided in engineering standards. Ultimately, this work provides a rather innovative and adaptable approach for structural integrity evaluations and critical engineering assessments through the proposal of an ANN enhanced FAD approach, simplifying these calculations while maintaining high fidelity.

Rapid review method series: interim guidance for the reporting of rapid reviews

Rapid reviews (RRs) are produced using abbreviated methods compared with standard systematic reviews (SR) to expedite the process for decision-making. This paper provides interim guidance to support the complete reporting...

Further results on alternating two-stage iterative method

Further results on alternating two-stage iterative method

Modified two-step modulus-based matrix splitting iteration methods for implicit complementarity problems

Modified two-step modulus-based matrix splitting iteration methods for implicit complementarity problems

Analysis of the Effect of Sampling Probe Geometry on Measurement Accuracy in Supersonic Gas Flow

The accuracy of sampling of gas components has a significant impact on the measurement of various performance parameters in the combustion chamber of an aero-engine. In order to investigate the effect of the probe geometry of a six-point gas sampling probe on sampling accuracy in supersonic gas flow, a three-dimensional probe gas flow characteristic solution model is established through numerical simulation methods of components of transport and fluid–solid coupling. Probes with three angles of 28°, 30°, and 32° and an optimized conical probe are constructed. The sampling accuracy of the probes with different geometries is compared and evaluated by the deviation of the component volume fraction before and after sampling and the resulting combustion efficiency error. This paper presents a set of calculation methods for solving the relative deviation of volume fraction by an iterative method based on the ideal gas law and the Redlich–Kwong equation (R-K equation). The method is designed to solve the exact component volume fraction problem in the simulation calculation. The study results demonstrate that the 28° and optimized conical probes improve sampling accuracy more effectively than the original 30° structure. The deviation of the volume fractions of the two structures is less than 1.7%, and the combustion efficiency error is less than 0.09%. The developed iterative calculation method can significantly reduce the theoretical calculation error to less than 0.06%. The experimental data of the test bench are in good agreement with the simulation results, thereby demonstrating the reliability and accuracy of the sampling probe following structural optimization.

Low-Bit-Depth Detection for Phase Retrieval with Higher Efficiency in Holographic Data Storage

In the past, comprehensive information was imperative for image processing, prompting a preference for high-depth cameras. However, in our research, we discovered that the abundance of image details may impede phase retrieval. Consequently, this paper presents an iterative phase retrieval method based on a low bit depth. Through simulations and experiments, this approach has proven effective in evidently enhancing phase retrieval outcomes. Furthermore, the concept of low bit depth holds promise for broader application across diverse domains within the field of image retrieval.

The acquisition and calibration of coarse-grained force field parameters for graphene/nitrile rubber nanocomposites

Graphene (GNS) is an ideal reinforcing material for developing high-performance nitrile rubber (NBR) nanocomposites to withstand complex and harsh working conditions. The mesoscale molecular simulation is significant in exploring the microscopic mechanism of nanocomposites. Initially, a coarse-grained (CG) scheme for NBR was established, and the potential parameter files were obtained through the iterative Boltzmann inversion method. Then, the bond and angle parameters were fitted by harmonic potential, the non-bonded interactions were fitted by LJ 12–6 potential. The CG force field parameters were coefficients of the potential function. The parameters such as the stiffness values, potential well parameter, etc., were calibrated by stretching the NBR molecular chains, and comparing the density and stretching performances of the NBR all-atomic and CG models. A rapid method for obtaining the non-bonded parameters between NBR and GNS was proposed based on the radial distribution function curves. The non-bonded parameters were calibrated by comparing GNS pull-out simulations, and the density and tensile properties of the GNS/NBR models. Finally, the CG force field parameters applicable to the GNS/NBR nanocomposites were obtained. The study is expected to be significant for subsequent mesoscale studies on GNS/NBR nanocomposites.

Acoustic Sensors for Monitoring and Localizing Partial Discharge Signals in Oil-Immersed Transformers under Array Configuration.

Partial discharge (PD) is one of the major causes of insulation accidents in oil-immersed transformers, generating a large number of signals that represent the health status of the transformer. In particular, acoustic signals can be detected by sensors to locate the source of the partial discharge. However, the array, type, and quantity of sensors play a crucial role in the research on the localization of partial discharge sources within transformers. Hence, this paper proposes a novel sensor array for the specific localization of PD sources using COMSOL Multiphysics software 6.1 to establish a three-dimensional model of the oil-immersed transformer and the different defect types of two-dimensional models. "Electric-force-acoustic" multiphysics field simulations were conducted to model ultrasonic signals of different types of PD by setting up detection points to collect acoustic signals at different types and temperatures instead of physical sensors. Subsequently, simulated waveforms and acoustic spatial distribution maps were acquired in the software. These simulation results were then combined with the time difference of arrival (TDOA) algorithm to solve a system of equations, ultimately yielding the position of the discharge source. Calculated positions were compared with the actual positions using an error iterative algorithm method, with an average spatial error about 1.3 cm, which falls within an acceptable range for fault diagnosis in transformers, validating the accuracy of the proposed method. Therefore, the presented sensor array and computational localization method offer a reliable theoretical basis for fault diagnosis techniques in transformers.

Multiple Solutions to the Fractional p-Laplacian Equations of Schrödinger–Hardy-Type Involving Concave–Convex Nonlinearities

This paper is concerned with nonlocal fractional p-Laplacian Schrödinger–Hardy-type equations involving concave–convex nonlinearities. The first aim is to demonstrate the L∞-bound for any possible weak solution to our problem. As far as we know, the global a priori bound for weak solutions to nonlinear elliptic problems involving a singular nonlinear term such as Hardy potentials has not been studied extensively. To overcome this, we utilize a truncated energy technique and the De Giorgi iteration method. As its application, we demonstrate that the problem above has at least two distinct nontrivial solutions by exploiting a variant of Ekeland’s variational principle and the classical mountain pass theorem as the key tools. Furthermore, we prove the existence of a sequence of infinitely many weak solutions that converges to zero in the L∞-norm. To derive this result, we employ the modified functional method and the dual fountain theorem.

Adaptive choice of near-optimal expansion points for interpolation-based structure-preserving model reduction

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative rational Krylov algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures.

A Robust Sparse Sensor Placement Strategy Based on Indicators of Noise for Ocean Monitoring

A well-performing data-driven sparse sensor deployment strategy is critical for marine monitoring systems, as it enables the optimal reconstruction of marine physical quantities with fewer sensors. However, ocean data typically contain substantial amounts of noise, including outliers (incomplete data) and inherent measurement noise, which heightens the complexity of sensor deployment. Therefore, this study optimizes the sparse sensor placement model by establishing noise indicators, including small noise weight and large noise weight, which are measured by entropy to minimize the prediction bias. Building on this, a robust sparse sensor placement algorithm is proposed, which utilizes the block coordinate update (BCU) iteration method to solve the function. During the iterative updating process, the proposed algorithm simultaneously updates the selection matrix, reconstruction matrix, and noise matrix. This allows for effective identification and mitigation of noise in the data through evaluation. Consequently, the deployed sensors achieve superior reconstruction performance compared to other deployment methods that do not incorporate noise evaluation. Experiments are also conducted on datasets of sea surface temperature (SST) and global ocean salinity, which demonstrate that our strategy significantly outperforms several other considered methods in terms of reconstruction accuracy while enabling autonomous sensor deployment under noisy conditions.

Distributed random swap: An efficient algorithm for minimum sum-of-squares clustering

The clustering model known as Minimum Sum-of-Squares Clustering (MSSC) is widely used, with the popular k-means algorithm serving as its local minimizer. It is well-known that solutions of k-means can result in substantial deviations from the true global optimum of MSSC. While numerous heuristics and metaheuristics have been proposed to overcome this limitation, none have gained dominant acceptance in academic literature. This is likely related to challenges such as intricate implementations and a multitude of tunable parameters.In this paper, we dispute the belief that simplifying an algorithm for MSSC inherently means sacrificing quality. We present the Distributed Random Swap (DRS-means) algorithm, which is designed to enhance clustering performance for the MSSC problem. This algorithm can be interpreted as an iterative method that refines the solution generated by the k-means algorithm during each iteration. The enhancement is achieved by selecting points based on specific probability distributions. These distributions are carefully designed to improve and speed up the exploration phase. The proposed algorithm is straightforward to implement. DRS-means offers a user-friendly solution with state-of-the-art results, making it suitable for a wide range of research fields.

Sparsity-driven loudspeaker gain optimization for sound field reconstruction with spherical microphone array

The paper presents a sparsity-driven method utilizing loudspeakers to reconstruct spatial sound fields using measurements obtained from a spherical microphone array (SMA). Employing spherical harmonics decomposition (SHD), the SMA recordings are characterized in the spherical harmonics domain. The gains for the loudspeakers are determined through an optimization problem, equating spherical harmonics pressure coefficients from primary and secondary sources. Furthermore, the sparsity within the loudspeaker feeds is redefined as a constrained sparse optimization problem, integrating linearity and orthogonality constraints. This method effectively reduces the required loudspeakers while maintaining sound field quality. The Bregman iteration method is applied to solve the constrained optimization problem. Rigorous evaluation based on reconstructed sound fields and objective measures highlights significant enhancements compared to least square and compressed sensing methods.

Some properties of the basins of attraction of the Newton’s method for simple nonlinear geodetic systems

Abstract The research presented in this paper concerns the determination of the attraction basins of Newton’s iterative method, which was used to solve the non-linear systems of observational equations associated with the geodetic measurements. The simple observation systems corresponding to the intersections or linear and angular resections used in practice were considered. The main goal was to investigate the properties of the sets of convergent initial points of the applied iterative method. Therefore, the answers to the questions regarding the geometric structure of the basins, their limitations, connectedness, or self-similarity were sought. The research also concerned the iterative structures of the basin: maps of the number of iterations which are necessary to achieve the convergence of the Newton’s method. The determined basins were compared with the areas of convergence that result from theorems on the convergence of the Newton’s method: the conditions imposed on the eigenvalues and norms of the matrices of the studied iterative systems. One of the significant results is the indication that the obtained basins of attraction contain areas resulting from the theoretical premises. Their diameters can be comparable with the sizes of the analyzed geodetic structures. Consequently, in the analyzed cases, it is possible to construct methods that enable quick selection of the initial starting points or automation of such selection. The paper also characterizes the global convergence mechanism of the Newton’s method for disconnected basins and, as a consequence, the non-local initial points located far from the solution points.

Theoretical Evaluation of Lubrication Performance of Thrust-Type Foil Bearings in Liquid Nitrogen

The development of reusable liquid rocket turbopumps has gradually highlighted the disadvantages of rolling bearings, particularly the contradiction between long service life and high rotational speed. It is critical to explore a feasible bearing scheme offering a long wear life and high stability to replace the existing rolling bearings. In this study, liquid nitrogen is adopted to simulate the ultra-low temperature environment of liquid rocket turbopumps, and theoretical evaluations of the lubrication performance of thrust-type foil bearings in liquid nitrogen are conducted. A link-spring model for the bump foil structure and a thin-plate finite element model for the top foil structure are established. The static and dynamic characteristics of the bearings are analyzed using methods including the finite difference method, the Newton–Raphson iteration method, and the finite element method. Detailed analysis includes the effects of factors such as rotational speed, fluid film thickness, thrust disk tilt angle, and the friction coefficient of the bump foil interface on the static and dynamic characteristics of thrust-type foil bearings. The research results indicate that thrust-type foil bearings have a good load-carrying capacity and low frictional power consumption. The adaptive deformation of the foil structure increases the fluid film thickness, preventing dry friction due to direct contact between the rotor journal and the bearing surface. When faced with thrust disk tilt, the direct translational stiffness and damping coefficient of the bearing do not undergo significant changes, ensuring system stability. Based on the results of this study, the exceptional performance characteristics of thrust-type foil bearings make them a promising alternative to rolling bearings for the development of reusable liquid rocket turbopumps.