Iterative Convergence Research Articles

A Branch-and-Cut-and-Price Algorithm for Cutting Stock and Related Problems

In this project, we introduce a branch-and-cut-and-price framework to solve the Cutting Stock Problems with strong relaxations using the Set Covering (Packing) Formulations, which are solved through column generation. We propose an extended Ryan-Foster branching scheme tailored to non-binary models, a pricing algorithm that produces convergence in a few iterations, and a variable selection technique based on branching history. These strategies are combined with subset-row cuts and custom primal heuristics to create a framework that overcomes the current state-of-the-art of Cutting Stock Problem, Skiving Stock Problem, and other related problems, being at least twice faster in the first problem and at least 60% faster in the second one.

Analyzing Convergence and Accuracy of Load Flow Methods in the IEEE 39-Bus Power System

This research paper presents a comprehensive comparative analysis of load flow methods applied to the IEEE 39 bus system, focusing on Newton-Raphson, Gauss-Seidel, and two variations of the Fast Decoupled method, XB and BX. The study evaluates the convergence behavior and iteration performance of these methods across two widely used software platforms: MATLAB and Power World Simulator. Through extensive simulations, the research investigates the convergence characteristics and computational efficiency of each method, providing insights into their respective advantages and limitations. The findings offer valuable guidance for power system engineers and researchers in selecting suitable load flow methods for analyzing complex power systems.

Innovative heat management method and metaheuristic algorithm optimized power supply-demand balance for PEMFC-ASHP-CHP system

The evolution of distributed building energy systems fuels the growing demand for sustainable energy solutions. In this paper, Proton Exchange Membrane Fuel Cell (PEMFC) and Air Source Heat Pump (ASHP) were integrated to form PEMFC-ASHP-CHP systems in three combination methods, i.e., Direct Combination (DC), Parallel Combination (PC), Series Combination (SC). And compared the energy management strategies and power balance of the system. To further improve reliability and flexibility, a diversion PEMFC-ASHP-CHP system was proposed by combining PC and SC's system advantages. Additionally, an iterative algorithm addressed the mismatch between power supply and demand. An empirical formula was proposed to improve iterative convergence speed for practical control situations. The coefficients were optimized using six metaheuristic algorithms, and the outcomes were summarized into an optimized operational plane to enhance the system control response speed further. The results show that the PC method performs better than DC and SC. It achieves a power consumption reduction of 52.8% and a COP improvement of 111.4% compared with the ASHP system. Meanwhile, the diversion system can more effectively utilize waste heat from the PEMFC and further improve the system's performance. The Queuing Search Algorithm (QSA) demonstrates superior accuracy for coefficient optimization. By using the established empirical formula, the convergence speeds of global and local iterative methods are improved by 26.10% and 41.78%, respectively, compared to the direct iterative algorithm. Ultimately, the optimized operational plane can achieve a maximum 37.16% hydrogen consumption reduction compared to the unoptimized system.

Metarouting with automatic tunneling in multilayer networks

Metarouting allows for the modeling of routing protocols using an algebraic structure called routing algebra. Routing protocols requiring design or validation can easily be modeled using this approach. To date, however, existing research on routing algebras has mainly focused on applying this approach to routing protocols that are generally used in networks which have a single addressing and forwarding protocol. The basic algebraic structures used in such contexts are semirings, Sobrinho’s algebras and algebras of endomorphisms. In this paper, we propose the modification of these existing routing algebras to deal with networks that contain multiple forwarding protocols where tunnels are omnipresent. To achieve this, we define new algebraic structures derived from the three aforementioned ones, in order to model the generalized routing problem with automatic tunneling entitled valid paths algebra. All of our routing algebras are defined as semi-direct products of two structures, the well-known shortest paths algebra and the proposed valid paths algebra. These new algebras are isotonic and non-monotonic with a partial order. We propose a fixed point for those new algebras and we prove the iterative convergence to the optimal solution of the valid shortest paths problem.

Non-stationary Anderson acceleration with optimized damping

Anderson acceleration (AA) has a long history of use and a strong recent interest due to its potential ability to dramatically improve the linear convergence of the fixed-point iteration. Most authors are simply using and analyzing the stationary version of Anderson acceleration (sAA) with a constant damping factor or without damping. Little attention has been paid to nonstationary algorithms. However, damping can be useful and is sometimes crucial for simulations in which the underlying fixed-point operator is not globally contractive. The role of this damping factor has not been fully understood. In the present work, we consider the non-stationary Anderson acceleration algorithm with optimized damping (AAoptD) in each iteration to further speed up linear and nonlinear iterations by applying one extra inexpensive optimization. We analyze the convergence rate this procedure and develop an efficient and inexpensive implementation scheme. We show by extensive numerical experiments that the proposed non-stationary Anderson acceleration with optimized damping procedure often converges much faster than stationary AA with constant damping, adaptive damping or without damping, especially in the cases larger window sizes are needed. We also observe that simple strategies like using constant damping factors and adaptive damping factors, sometimes, work very well for some problems while sometimes they are even worse than AA without damping. Our proposed method is usually more robust than AA with constant damping and adaptive damping. Moreover, we also observed from our numerical results that damping can be good, but choosing the wrong damping factors may slow down the convergence rate. Theoretical analysis of the effects of damping factors are needed and important.

Application of Particle Swarm Optimisation in Multi-Objective Cost Optimisation of Engineering Enterprises under the Background of Digital Economy

Engineering projects must meet quality and schedule requirements during construction. This is a typical multi-objective problem and a difficult point in the management of engineering enterprises. To address these issues, a research study proposes an intelligent multi-objective optimisation technique. First, analyse the optimisation objectives of the enterprise in the context of digitalisation. Then, construct a multi-objective cost optimisation model for engineering enterprises. Second, the Multi-Objective Particle Swarm Optimisation (MOPSO) algorithm is introduced to solve multi- objective problems. To improve the multi-objective optimisation effect of the model, the inertia weight parameters and particle learning behaviour are optimised and adjusted, as the model is prone to getting stuck in local optima. In the performance test of the algorithm model, the optimised MOPSO model can accurately search for the minimum value of 0 at the position (0, 0) under the Rastrig in function, and at the same time, the number of iteration convergence is the least. The GA, ACOM, and traditional MOPSO models have more iterative convergence times, and the optimisation results are 0.10, 0.15, and 0.14, respectively. It can be seen that the performance of the optimised MOPSO model is better. In the specific example analysis, using the optimised MOPSO solution, the project cost was reduced from 31 million yuan in the contract to 30.52 million yuan, and the construction period was shortened from 588 days to 540 days, and met the environmental protection and quality requirements. The research content can provide important decision support for engineering project managers.

Quasi-Newton positivity-preserving scheme for the Spalart-Allmaras turbulence model using unstructured grids

A new implicit method for solving the Spalart-Allmaras (SA) turbulence model is proposed using a formal second-order upwind scheme on unstructured grids for steady-state flows. The mean-flow set of equations and the SA model equation are solved in a loosely coupled manner. While the Jacobian of the mean-flow equations is derived from a low-order approximation of the residual (i.e. the defect-correction approach), the SA Jacobian is obtained by modifying a direct linearization of the second-order residual. The modified Jacobian is designed to form an M-matrix without the artificial time derivative commonly used for steady-state problems. Namely, no time step is involved in solving the SA model equation. Thanks to a special design of the M-matrix, the positivity of the SA model working variable and the convergence of the SA model linearized problem (fixed-point iterative problem) is guaranteed. To further enhance the overall stability of the flow solver, the Runge-Kutta implicit algorithm is employed, allowing CFL numbers as high as one to two thousand for the mean-flow equations. A second-order upwind scheme is achieved with a least-square method using a limiter to suppress oscillation in the solution. However, second-order upwind scheme discretization of turbulence models may lack robustness, especially when using unstructured grids. The limiter can hinder the convergence of the non-linear iterations, especially of the turbulence model. A square root transformation is applied to the dependent variable of a baseline SA model to improve the non-linear convergence characteristics. Three variants of the SA model are implemented and tested together with two limiters. Numerical simulations of a broad range of aerodynamic applications are conducted. The transformed variant exhibits the least sensitivity to the limiter type.

Existence of a Fixed Point and Convergence of Iterates for Self-Mappings of Metric Spaces with Graphs

Existence of a Fixed Point and Convergence of Iterates for Self-Mappings of Metric Spaces with Graphs

Self-Calibration for Star Sensors.

Aiming to address the chicken-and-egg problem in star identification and the intrinsic parameter determination processes of on-orbit star sensors, this study proposes an on-orbit self-calibration method for star sensors that does not depend on star identification. First, the self-calibration equations of a star sensor are derived based on the invariance of the interstar angle of a star pair between image frames, without any requirements for the true value of the interstar angle of the star pair. Then, a constant constraint of the optical path from the star spot to the center of the star sensor optical system is defined to reduce the biased estimation in self-calibration. Finally, a scaled nonlinear least square method is developed to solve the self-calibration equations, thus accelerating iteration convergence. Our simulation and analysis results show that the bias of the focal length estimation in on-orbit self-calibration with a constraint is two orders of magnitude smaller than that in on-orbit self-calibration without a constraint. In addition, it is shown that convergence can be achieved in 10 iterations when the scaled nonlinear least square method is used to solve the self-calibration equations. The calibrated intrinsic parameters obtained by the proposed method can be directly used in traditional star map identification methods.

On the Poisson-transmuted exponential distribution and its application to frequency of claim in actuarial science

This study proposes a new discrete distribution in the mixed Poisson paradigm to obtain a distribution that provides a better fit to skewed and dispersed count observation with excess zero. The cubic transmutation map is used to extend the exponential distribution, and the obtained continuous distribution is assumed for the parameter of the Poisson distribution. Various moment-based properties of the new distribution are obtained. The Nelder-Mead algorithm provides the fastest convergence iteration under the maximum likelihood estimation technique. The shapes of the proposed new discrete distribution are similar to those of the mixing distribution. Frequencies of insurance claims from different countries are used to assess the performance of the new proposition (and its zero-inflated form). Results show that the new distribution outperforms other competing ones in most cases. It is also revealed that the natural form of the new distribution outperforms its zeroinflated version in many cases despite having observations with excess zero counts.

Optimization and performance enhancement of renewable energy microgrid energy system using pheasant bird optimization algorithm

The main aim of this research work is to design and select an optimal renewable energy resource based microgrid (MG) system for rural area electrification of India. MG system consists of diesel generator (DG), battery bank (BB), wind turbine (WT), biomass generator (BMG) and solar photovoltaic (PV) modules with seven different configurations. The optimal system is selected based on technological, economic, environmental, social and reliability parameters. For this objective, a novel algorithm called pheasant bird optimization algorithm (PBOA) is presented and compared with particle swarm optimization (PSO), genetic algorithm (GA), ant colony optimization (ACO), cuckoo search algorithm (CSA) and HOMER Pro. Results show that PV/WT/BMG/BB/DG is an optimal configuration with sizes of 1148 units of PV, 42 units of WT, 20 units of BMG, 36 units of BB and 1 unit of DG. PBOA has determined the optimal resource size for the MG system. In terms of social and reliability parameters for the PV/WT/BMG/BB/DG system, PBOA generated values of 93.58 %, for RF, 0.08754 for EGJC and 0.01 % for LPSP. For the optimal configuration, PBOA estimated NPC, COE, CE values are ₹66185779, ₹9.3 and 650901 kg/yrs respectively. PBOA reaches the minimal NPC of ₹66185779 in just 180 iterations, whereas PSO, ACO, GA, and CSA required 360, 200, 360 and 250 iterations respectively. PBOA also achieved minimal COE at a rate of 1.52, 1.11, 1.11 and 1.94 times faster than PSO, GA, CSA and ACO respectively. The contribution of PV and WT sources to the total generated electricity is approximately 50 % and 35 % respectively. PBOA exhibits faster convergence time and iteration compared to the PSO, ACO, GA and CSA algorithm. The comparison results indicate that the optimal PV/WT/BMG/BB/DG configuration select for rural areas. This research provides valuable information for policy makers and investors on the implementation of MG system in rural regions.

Fast simulation of airfoil flow field via deep neural network

Computational Fluid Dynamics (CFD) has become an indispensable tool in the optimization design, and evaluation of aircraft aerodynamics. However, solving the Navier-Stokes (NS) equations is a time-consuming, memory demanding and computationally expensive task. Artificial intelligence offers a promising avenue for flow field solving. In this work, we propose a novel deep learning framework for rapidly reconstructing airfoil flow fields. Channel attention and spatial attention modules are utilized in the downsampling stage of the UNet to enhance the feature learning capabilities of the deep learning model. Additionally, Embedding the predicted values of the deep learning model as initial values into the CFD solver to accelerate its iterative convergence. The NACA series airfoils were used to validate the prediction accuracy and generalization of the deep learning model. The experimental results represent the deep learning model achieving flow field prediction speed three orders of magnitude faster than CFD solver. Furthermore, the CFD solver integrated with deep learning model demonstrates a threefold acceleration compared to CFD solver. By extensively mining historical flow field data, an efficient solution is derived for the rapid simulation of aircraft flow fields.

High order compact augmented methods for Stokes equations with different boundary conditions

This paper is devoted to fourth order compact schemes and fast algorithms for solving stationary Stokes equations with different boundary conditions numerically. One of the main ideas is to decouple the Stokes equations into three Poisson equations for the pressure and the velocity via the pressure Poisson equation (PPE). The augmented strategy is utilized to provide numerical boundary conditions for the pressure. Different velocity boundary conditions require different interpolation strategies for the augmented methods. The augmented variable is solved by the GMRES method. A new simple and efficient preconditioning strategy has also been developed to accelerate the convergence of the GMRES iteration. Numerical examples presented in this paper confirmed the designed convergence order and the efficiency of the new methods.

Enhanced Moving Source Localization with Time and Frequency Difference of Arrival: Motion-Assisted Method for Sub-Dimensional Sensor Networks

Localizing a moving source by Time Difference of Arrival (TDOA) and Frequency Difference of Arrival (FDOA) commonly requires at least N+1 sensors in N-dimensional space to obtain more than N pairs of TDOAs and FDOAs, thereby establishing more than 2N equations to solve for 2N unknowns. However, if there are insufficient sensors, the localization problem will become underdetermined, leading to non-unique solutions or inaccuracies in the minimum norm solution. This paper proposes a localization method using TDOAs and FDOAs while incorporating the motion model. The motion between the source and sensors increases the equivalent length of the baseline, thereby improving observability even when using the minimum number of sensors. The problem is formulated as a Maximum Likelihood Estimation (MLE) and solved through Gauss–Newton (GN) iteration. Since GN requires an initialization close to the true value, the MLE is transformed into a semidefinite programming problem using Semidefinite Relaxation (SDR) technology, while SDR results in a suboptimal estimate, it is sufficient as an initialization to guarantee the convergence of GN iteration. The proposed method is analytically shown to reach the Cramér–Rao Lower Bound (CRLB) accuracy under mild noise conditions. Simulation results confirm that it achieves CRLB-level performance when the number of sensors is lower than N+1, thereby corroborating the theoretical analysis.

Iterative solution methods for high-order/hp–DGFEM approximation of the linear Boltzmann transport equation

In this article we consider the iterative solution of the linear system of equations arising from the discretisation of the poly-energetic linear Boltzmann transport equation using a high-order/hp–version discontinuous Galerkin finite element approximation in space, angle, and energy. In particular, we develop preconditioned Richardson iterations which may be understood as generalisations of source iteration in the mono-energetic setting, and derive computable a posteriori bounds for the solver error incurred due to inexact linear algebra, measured in a relevant problem-specific norm. We prove that the convergence of the resulting schemes and a posteriori solver error estimates are independent of the mesh size h and polynomial degree p. We also discuss how the poly-energetic Richardson iteration may be employed as a preconditioner for the generalised minimal residual (GMRES) method. Furthermore, we show that standard implementations of GMRES based on minimising the Euclidean norm of the residual vector can be utilized to yield computable a posteriori solver error estimates at each iteration, through judicious selections of left- and right-preconditioners for the original linear system. The effectiveness of poly-energetic source iteration and preconditioned GMRES, as well as their respective a posteriori solver error estimates, is demonstrated through numerical examples arising in the modelling of photon transport. While the convergence of poly-energetic source iteration is independent of h and p, we observe that the number of iterations required to attain convergence when employing GMRES only depends mildly on h and p. Moreover, this latter approach is highly effective in the low energy regime.

Performance-optimisation-based repetitive trajectory tracking control for autonomous farming vehicle

The high precision trajectory tracking for the autonomous farming vehicle (AFV) is closely related to work quality and crop yield. Farmland operations are usually repetitive tillages, and the farmland soil is relatively soft, slippery, and uneven, which can easily lead to uncertain problems such as slippage, parameter perturbation, and external disturbance. This paper proposes a finite-time repetitive trajectory tracking control strategy integrated with particle swarm optimisation (PSO) and equivalent input disturbance (EID) for the AFV to achieve performance optimisation. The repetitive control framework with finite-time convergence technique can effectively realise the iterative convergence performance of the repetitive trajectory tracking. The PSO algorithm is integrated into determining the control gains to optimise the dynamic and steady-state performances of the trajectory tracking control system. The EID method enhances the tracking precision and robustness to internal and external disturbances. Under MATLAB/Simulink and CarSim co-simulation environment, the effectiveness and advantages of the designed control strategy are illustrated by comparing it with the traditional repetitive control, the EID-based PI control, the active-disturbance-rejection-control-based nonsingular terminal sliding mode control, as well as the sliding-mode-observer-based integral sliding mode control strategies for a farming vehicle.

Mastering the art: Proficient finite element implementation and robust evaluation of a strain-hardening porous ductile material crack growth prediction model at finite strain

Accurate prediction of crack extension is crucial for ensuring the structural integrity of metal components exposed to diverse loads. While the Gurson model and its extensions are widely accepted for delineating ductile fracture stages, especially in porous materials with a rigid-perfectly plastic matrix, their efficacy diminishes in the presence of strain-hardening matrices. To address this limitation, our study introduces a robust numerical implementation and assessment of Perrin's model. In contrast to Gurson's approach, Perrin's model incorporates two distinct strain-hardening parameters derived from an intricate analysis of a strain-hardening hollow sphere subjected to axisymmetric loading.Moving beyond theoretical discussions, this paper actively engages in the rigorous evaluation of Perrin's model under various scenarios, encompassing isotropic, kinematic, and mixed isotropic-kinematic hardening conditions. The model's effectiveness is convincingly demonstrated through meticulous comparisons between numerical simulations and real-world experimental observations conducted on pre-cracked specimens.Facing challenges related to achieving global elastic-plastic convergence in large-scale simulations, our research introduces a sophisticated approach. Leveraging stiffness tangent moduli ensure the maintenance of quadratic convergence in global Newton method iterations. This not only enhances the reliability of our simulations but also underscores the practical applicability of our findings.The study also focuses on the necking and crack propagation of a cylindrical specimen through meticulous computational modeling. Our mesh captures intricate details and addresses stress gradients, revealing coplanar crack propagation contrary to prior beliefs. The confirmation of the cup-cone fracture effect challenges previous interpretations, emphasizing the critical role of void nucleation. This reinforces the credibility and applicability of our approach in advancing the understanding of material fracture behavior.In summary, this study significantly contributes to existing knowledge in fracture prediction models, offering a robust framework for predicting ductile fracture under substantial deformations. The presented findings pave the way for advancements in structural engineering, providing invaluable insights for optimizing the performance and resilience of metal structures in real-world applications.

Coupling of Eddy-Viscosity-Scale Reynolds-Stress Model with an Algebraic Transition Model

Attempts to extend a Reynolds-stress model (RSM) for transition flow simulations have begun to cause concern in recent years based on the fact that the RSM has more potential than an eddy viscosity model when applied to complex three-dimensional flowfields. In terms of the RSM, however, coupling more transport equations will further increase its complexity (may worsen robustness and convergence rate). In this paper, a γ-based algebraic transition model originating from the SA-BC model is coupled with an νt-scale RSM (EV-RSM) to form a new transition/turbulence model, called tEV-RSM. Some numerical tests are conducted with a fifth-order scheme, WCNS, including the well-known T3 flat plate series, a two-dimensional airfoil (S809), a two-dimensional three-element airfoil (30P-30N), a three-dimensional prolate-spheroid, and a three-dimensional circular cylinder. The proposed model is compared with the SA-BC and γ-Reθt SST models, which indicates that it can predict the flows primarily governed by Tollmien–Schlichting transition or bypass transition well and has an advantage in predicting the flows with detached separation. Meanwhile, the iteration convergence rate of the tEV-RSM is fast enough considering it as a seven-equation model.

A Globally Convergent Iterative Method for Matrix Sign Function and Its Application for Determining the Eigenvalues of a Matrix Pencil

In this research article, we propose a new matrix iterative method with a convergence order of five for computing the sign of a complex matrix by examining the different patterns and symmetry of existing methods. Analysis of the convergence of the method was explored on a global scale, and attraction basins were demonstrated. In addition to this, the asymptotic stability of the scheme was explored.Then, an algorithm for determing thegeneralized eigenvalues for the case of regular matrix pencils was investigated using the matrix sign computation. We performed a series of numerical experiments using numerous matrices to confirm the usefulness and superiority of the proposed method.

Effect of bending deformation on the electronic and optical properties of O atoms adsorbed by Be3N2.

In this paper, the optimum coverage of 4.44% and the optimum adsorption sites were determined for the Be3N2 adsorption system of O atoms at different coverages based on density functional theory. The electronic and optical properties of the model were investigated by applying bending deformation to the model at these coverage and adsorption sites. Adsorption of O atoms disrupts the geometrical symmetry of Be3N2, resulting in orbital rehybridization and lowering its band gap. Bending deformation causes the band gap of the adsorbed O atom structure of Be3N2 to first increase and then decrease, resulting in the modulation of its band gap. With increasing bending deformation, the adsorbed system is redshifts, and the degree of redshift increases with increasing bending deformation. All calculations in this paper were performed using the first-principles-based CASTEP module of Materials Studio (MS). The generalized gradient approximation (GGA) plane-wave pseudopotential method and the Perdew-Burke-Ernzerhof (PBE) Perdew et al. Phys Rev Lett 77:3865, 1996 generalized functional were used in the geometry optimization and calculation process to calculate the exchange-correlation potential between electrons. The effect of coverage on the electronic and optical properties of the Be3N2-adsorbed O atom system was investigated by adsorbing different numbers of O atoms on a monolayer of Be3N2. The Be3N2 protocell contains two N atoms and three Be atoms with a space community of P6/MMM (No.191). The original cell was expanded 3 times along the direction of the base vectors a and b in the Be3N2 plane to create a 3 × 3 × 1 monolayer Be3N2 supercell system. A vacuum layer of 15Å is set in the direction of the crystal plane of the vertical monolayer Be3N2 supercell to eliminate interactions between adjacent layers. In the overall energy convergence test of the Be3N2 supercell, the plane wave truncation energy was set to 500eV, and the energy difference between the calculations given in the literature Reyes-Serrato et al. J Phys Chem Solids 59:743-6, 1998 using 550eV was less than 0.01eV, verifying the reliability of the data at a truncation energy of 500eV. The Monkhorst-Pack special k-point sampling method Monkhorst et al. Phys Rev B 13:5188, 1976 was used in the structural calculations, and the grid was set to 3 × 3 × 1. The geometric optimization parameters are set as follows: the self-consistent field iteration convergence criterion is 2.0 × 10-6eV, and the iterative accuracy convergence value is not less than 1.0 × 10-5eV/atom for the total force of each atom and less than 0.03eV/Å for all atomic forces. In addition the high-symmetry k-point path is taken as Γ(0,0,0) → M(0,0.5,0) → K(- 1/3,2/3,0) → Γ(0,0,0) Chen et al, AIP Adv 8:105105, 2018.