Suitable Solver Research Articles

A high-order finite-difference solver for direct numerical simulations of magnetohydrodynamic turbulence

This paper presents the development and validation of a Magnetohydrodynamics (MHD) module integrated into the Xcompact3d framework, an open-source high-order finite-difference suite of solvers designed to study turbulent flows on supercomputers. Leveraging the Fast Fourier Transform library already implemented in Xcompact3d, alongside sixth-order compact finite-difference schemes and a direct spectral Poisson solver, both the induction and potential-based MHD equations can be efficiently solved at scale on CPU-based supercomputers for fluids with strong and weak magnetic field, respectively. Validation of the MHD solver is conducted against established benchmarks, including Orszag-Tang vortex and MHD channel flows, demonstrating the module's capability to accurately capture complex MHD phenomena, providing a powerful tool for research in both engineering and astrophysics. The scalability of the Xcompact3d framework remains intact with the incorporation of the MHD module, ensuring efficient performance on modern high-performance clusters. This paper also presents new findings on the evolution of the Taylor-Green vortex under an external magnetic field for different flow regimes.PROGRAM SUMMARYProgram Title: Xcompact3dCPC Library link to program files:https://doi.org/10.17632/z835zbbs8g.1Developer's repository link:https://github.com/Xcompact3d/Xcompact3dLicensing provisions: BSD 3-Clause LicenseProgramming language: FortranNature of problem: Magnetohydrodynamics.Solution method: High-order finite-difference method and spectral based Poisson solver.Additional comments including restrictions and unusual features: Heterogeneous parallel capabilities with GPUs are under active development.

A new shifted generalized Chebyshev approach for multi-dimensional sinh-Gordon equation

The numerical treatment of multi-dimensional non-linear sinh-Gordon equations is the focus of this paper. We numerically solve the (1 + 1) and (2 + 1) sinh-Gordon equations using two collocation algorithms. We select the set of basis functions as a set of generalized Chebyshev polynomials (CPs), which we express as orthogonal combinations of CPs. We develop and utilize some formulas related to these polynomials to propose our numerical algorithms. Specific values for the high-order derivatives of the basis functions serve in the derivation of the two presented algorithms. Additionally, we provide an estimation of the basis functions used in the convergence analysis study. We follow the two collocation algorithms to transform the sinh-Gordon equations into non-linear equation systems, which any suitable solver can handle. We provide some examples and comparisons to confirm the effectiveness of our presented algorithms.

Demand Response Method for the Domestic PV system

This paper describes the deployment of the well-known Demand Response principle in a domestic PV inverter and spot price tariff environment to minimize power purchase costs. The optimization problem is defined as a pure linear program which is solved using a suitable solver. The results are verified using a dataset that was obtained by metering at a real storage site that was managed by an expert on each monitored day.

Comparative Study of OpenFOAM Solvers on Separation Pattern and Separation Pattern Transition in Overexpanded Single Expansion Ramp Nozzle

Flow separation in overexpanded single expansion ramp nozzles (SERN) involves complex phenomena, such as shock waves, expansion waves, turbulent boundary layers, and shear layers. Computational fluid dynamics plays a crucial role in studying unsteady flow behaviour in supersonic nozzles, allowing for an investigation into the dynamic flow field characteristics. However, the application of OpenFOAM as a numerical tool for studying SERN in the field of compressible flows, particularly in the overexpansion state where the flow field characteristics are more complex, has received relatively less attention. In this study, the flow field characteristics of an overexpanded SERN under different turbulence models are investigated through a combination of experiments and numerical calculations. The qualitative and quantitative predictive performance of two compressible flow solvers in OpenFOAM, namely, rhoCentralFOAM and sonicFOAM, are compared in terms of flow separation pattern and separation pattern transitions within the overexpanded SERN. The ability of rhoCentralFOAM and sonicFOAM to accurately predict complex flow states is evaluated. Results indicate that the numerical simulations conducted using rhoCentralFOAM and sonicFOAM successfully capture flow separation, separated shock waves, separated bubbles and shear layers for two types of restricted shock separation patterns at the same nozzle pressure ratio (NPR), demonstrating agreement with experimental results. However, sonicFOAM initiates the transition in the separation pattern 0.0773 NPR earlier than rhoCentralFOAM during the whole separation pattern transition process of the SERN. The transition process in sonicFOAM lasts longer and exhibits a greater variation in NPR. SonicFOAM fails to accurately predict certain aspects, such as the pressure rise after the separation bubble, the reattachment shock wave, and tends to overestimat the length of the separation shock length. Consequently, sonicFOAM cannot be recommended as a suitable solver for accurately capturing the separation pattern of an overexpanded nozzle.

Numerical Solutions of High-Order Differential Equations with Polynomial Coefficients Using a Bernstein Polynomial Basis

The paper presents a novel method that allows one to establish numerical solutions of linear and nonlinear ordinary differential equations—with polynomial coefficients—that contain any finite products of the unknown functions and/or their general derivatives. The presented algorithm provides numerical solutions of these differential equations subject to initial or boundary conditions. This algorithm proposes the desired solution in terms of B-polynomials (Bernstein polynomial basis) and then uses the orthonormal relation of B-polynomials with its weighted dual basis with respect to the Jacobi weight function to construct a linear/nonlinear system in the unknown expansion coefficients which can be solved using a suitable solver. The properties of B-polynomials provide greater flexibility in which to impose the initial or boundary conditions at the end points of the interval [0, R] and enable us to obtain exactly and explicitly some of the unknown expansion coefficients in the form of a suggested numerical solution. Consequently, the presented algorithm leads to a linear or nonlinear algebraic system in the unknown expansion coefficients that has a simpler form than that was obtained by the other algorithms. So that, this procedure is a powerful tool that we may utilize to overcome the difficulties associated with boundary and initial value problems with less computational effort than the other techniques. An accepted agreement is obtained between the exact and approximate solutions for the given examples. The error analysis was also studied, and the obtained numerical results clarified the validity of the theoretical results.

The role of risk-based demand response in resource management of a grid-connected renewable-based large-scale microgrid with stationary and mobile energy storage systems and emission tax

Integration of energy storage systems with local electricity systems, like microgrids, is emerging due to their economical, reliability, and environmental benefits. These advantages can be achieved by load management subject to the related condition, such as the hour of the day, electricity price, etc. From an economic point of view, energy storage systems can reduce the share of costly power generators in supplying the peak demand. In other words, they can be charged in low-demand hours and discharged in peak-demand hours, when the electricity price is high. In this work, the impact of stationary battery storage and electric vehicles on the resource management of a large-scale microgrid is assessed through a stochastic model. The understudy microgrid includes also renewable energy sources (namely solar photovoltaic systems and wind turbines) as well as diesel generators. This study considers the uncertainty in the electricity price, where a scenario-based risk management approach is employed to control the risk associated with price uncertainty. The resulting model is implemented in the GAMS software package to be solved by a suitable solver. Numerical simulation of a large-scale microgrid is accomplished to validate the effectiveness of energy storage systems in the economic improvement of the understudy microgrid.

A Feature-Independent Hyper-Heuristic Approach for Solving the Knapsack Problem

Recent years have witnessed a growing interest in automatic learning mechanisms and applications. The concept of hyper-heuristics, algorithms that either select among existing algorithms or generate new ones, holds high relevance in this matter. Current research suggests that, under certain circumstances, hyper-heuristics outperform single heuristics when evaluated in isolation. When hyper-heuristics are selected among existing algorithms, they map problem states into suitable solvers. Unfortunately, identifying the features that accurately describe the problem state—and thus allow for a proper mapping—requires plenty of domain-specific knowledge, which is not always available. This work proposes a simple yet effective hyper-heuristic model that does not rely on problem features to produce such a mapping. The model defines a fixed sequence of heuristics that improves the solving process of knapsack problems. This research comprises an analysis of feature-independent hyper-heuristic performance under different learning conditions and different problem sets.

Wind Environment Simulation Accuracy in Traditional Villages with Complex Layouts Based on CFD.

Using wind speed, wind direction, and turbulence intensity values as evaluation indicators, the ventilation performance of villages with complex building layouts was studied. We used the SKE, RNG, and RKE solvers in CFD-3D steady-state Reynolds-averaged Navier–Stokes (RANS) to simulate the wind environment of a village. The findings show that for the simulation of rural wind environments with complex building layouts, steady-state simulation solvers need to be evaluated in detail to verify their accuracy. In this study, a village with a complex architectural layout in Southern Shaanxi, China, was taken as the research object, and three steady-state simulation solvers were used to evaluate the ventilation performance of the village. The simulated data were compared with the measured data to find the most suitable solver for this kind of village wind environment simulation. The results show that for the simulation of the village wind environment with a complex building layout, the RNG simulation results have the lowest reliability among the three steady-state solvers. The reliability of wind speed distribution and turbulence intensity distribution are 0.7881 and 0.2473, respectively. However, the wind speed and turbulence intensity values obtained by the SKE solver are the closest to the measured values, which are 0.8625 and 0.9088, respectively. Therefore, for villages with complex building layouts, the SKE solver should be the first choice for simulating wind environment distribution. When using the RNG solver, the overall turbulence intensity value obtained is higher than the measured value. The average deviation between the simulated data and SKE and RKE at a height of 1.7 m is 42.61%. The main reason for this is that RNG overestimates the vortices and underestimates the airflow rate in the building intervals.

Numerical study of surface thermal signatures of lee waves excited by moving underwater sphere at low Froude number

A submerged body moving in density-stratified water, such as in oceans, disturbs the surrounding water layers. Furthermore, the internal waves generate and modulate the flow field on the water surface. Thereafter, the thin water surface layer and underlying water, which differ in temperatures because of factors such as solar radiation and water vapor exchange, mix. Consequently, thermal signatures could be generated on the water surface and detected by high-resolution infrared equipment; hence, the presence of a submerged moving body could also be deduced. To thoroughly examine this phenomenon, a submerged sphere moving horizontally in linearly density-stratified water with a “hot skin” layer on its surface was numerically studied in this work. A suitable solver, i.e., “twoLiquidMixing_skin_Foam”, based on the well-known open-source computational fluid dynamics code, “Open-Source Field Operation and Manipulation”, was developed for the aforementioned problem. The internal waves excited by the sphere were validated by experimental results to illustrate the accuracy of the developed solver. Then, the impact of lee waves on the water surface thermal signatures was investigated in detail for the towed submerged sphere moving at various depths in fluid with different internal Froude numbers.

SolveSAPHE-r2 (v2.0.1): revisiting and extending the Solver Suite for Alkalinity-PH Equations for usage with CO<sub>2</sub>, HCO<sub>3</sub><sup>−</sup> or CO<sub>3</sub><sup>2−</sup> input data

Abstract. The successful and efficient approach at the basis of the Solver Suite for Alkalinity-PH Equations (SolveSAPHE) (Munhoven, 2013), which determines the carbonate system speciation by calculating pH from total alkalinity (AlkT) and dissolved inorganic carbon (CT), and which converges for any physically sensible pair of such data, has been adapted and further developed to work with AlkT–CO2, AlkT–HCO3-, and AlkT–CO32-. The mathematical properties of the three modified alkalinity–pH equations are explored. It is shown that the AlkT–CO2, and AlkT–HCO3- problems have one and only one positive root for any physically sensible pair of data (i.e. such that [CO2]>0 and [HCO3-]>0). The space of AlkT–CO32- pairs is partitioned into regions where there is either no solution, one solution or where there are two. The numerical solution of the modified alkalinity–pH equations is far more demanding than that for the original AlkT–CT pair as they exhibit strong gradients and are not always monotonous. The two main algorithms used in SolveSAPHE v1 have been revised in depth to reliably process the three additional data input pairs. The AlkT–CO2 pair is numerically the most challenging. With the Newton–Raphson-based solver, it takes about 5 times as long to solve as the companion AlkT–CT pair; the AlkT–CO32- pair requires on average about 4 times as much time as the AlkT–CT pair. All in all, the secant-based solver offers the best performance. It outperforms the Newton–Raphson-based one by up to a factor of 4 in terms of average numbers of iterations and execution time and yet reaches equation residuals that are up to 7 orders of magnitude lower. Just like the pH solvers from the v1 series, SolveSAPHE-r2 includes automatic root bracketing and efficient initialisation schemes for the iterative solvers. For AlkT–CO32- data pairs, it also determines the number of roots and calculates non-overlapping bracketing intervals. An open-source reference implementation of the new algorithms in Fortran 90 is made publicly available for usage under the GNU Lesser General Public Licence version 3 (LGPLv3) or later.

Multi-objective particle swarm optimization and training of datasheet-based load dependent lithium-ion voltage models

In this study, a datasheet-based lithium-ion (Li-ion) battery cell model is built, trained and validated by utilizing Multi-Objective Particle Swarm Optimization (MOPSO). Firstly, in order to determine the most suitable solver among various multi-objective solutions, a first-order (1RC) Thevenin model is adopted. At this stage, a thorough comparison is carried out among the Mean Squared Errors (MSE), Mean Signed (MSD) and Average Deviations (AD) of five study cases, where the proposed solver showed both up to 71%,96% and 54% improvements respectively compared to the previously reported optimal result, and capability to converge over the whole available capacity of the cells. Secondly, the optimal solver selection is followed by a comparative study among various equivalent circuit models (1RC, 2RC and 3RC), which are thoroughly evaluated upon computational time, model complexity and accuracy. Thirdly, the proposed optimized Thevenin Equivalent Circuit Model (ECM) is parameterized and compared to an experiment-based model of a commercial high-power Li-ion cell, for both charge and discharge processes. Outcomes indicate an improved performance of the proposed methodology which is ultimately utilized to train a dataset obtained from the manufacturer and successfully apply it on several unspecified current profiles. As a result, a wide range of accurate voltage-model outputs is acquired from a few input profiles, indicating the overall robustness of the method and rendering it as a strong candidate for non-dynamic loading battery applications.

Dynamic robust stabilization of fractional-order linear systems with nonlinear uncertain parameters: an LMI approach

ABSTRACT This paper presents a dynamic output feedback controller with determined order for the stabilization of a class of fractional-order system with nonlinear uncertain parameters with fractional order . Using stability theories of fractional-order systems and linear matrix inequalities (LMIs), some sufficient conditions in the LMI form are deduced to guarantee the robustness and asymptotic stabilization of the system. Designing a dynamic robust controller, along with all its useful features, leads to more unknown parameters in comparison with a static controller and makes controller design procedure more difficult due to more complex constraints that must be solved. In this paper, using proper lemmas and theorems, LMI techniques, and suitable solvers and parsers the difficulty of designing such controllers has been overcome. Simulation results of three different numerical examples illustrate that the proposed sufficient theoretical results are applicable and effective for tackling robust stabilization problems.

Generating Cryptographic S-Boxes Using the Reinforcement Learning

Substitution boxes (S-boxes) are essential components of many cryptographic primitives. The Dijkstra algorithm, SAT solvers, and heuristic methods have been used to find bitsliced implementations of S-boxes. However, it is difficult to apply these methods for 8-bit S-boxes because of their size. Therefore, to implement these S-boxes so that the countermeasure of side-channel attack can be applied efficiently, using structures such as Feistel, Lai-Massey, and MISTY that can be bitsliced implemented with a small number of nonlinear operations has been widely used. Since S-boxes constructed with structures consist of small S-boxes and have specific designs, there are limitations to their cryptographic security and efficiency. In this paper, we propose a new method for generating S-boxes by stacking bitwise operations from the identity function, an approach that is different from existing methods. This method can be expressed in Markov decision process, and reinforcement learning is a suitable solver for Markov decision process. Our goal is to train this method to an agent through reinforcement learning to generate S-boxes to which the masking scheme, which is a countermeasure of side-channel attack, can be efficiently applied. In particular, our method provided various S-boxes superior or comparable to existing S-boxes. We produced 8-bit S-boxes with differential uniformity 16 (resp. 32) and linearity 128 (resp. 128), generated with nine (resp. eight) nonlinear operations, for the first time. To our best knowledge, this is the first study to construct cryptographic S-Box by incorporating reinforcement learning.

Assessment of CFD Model Performance for Flows around a Hydraulic Structure of Complex Geometry

Computational Fluid Dynamics (CFD) modeling has become a widely used research tool in the hydraulic engineering community, however, for many problems the numerical model setup and even the selection of the suitable hydrodynamic solver is still a challenging task. This is the case for fine scale analysis of flow features around hydraulic structures of complex geometry, where generation of the 3D grid, application of the correct turbulence modeling approach, or modeling of the free surface all require extensive experience and knowledge. The aim of this paper is to perform a numerical hydrodynamic experiment for a hydraulic structure with complex geometry to assess the model performance, in terms of grid resolution, grid refinement methods as well as turbulence modeling. The open source modeling environment of OpenFOAM is tested and validated against laboratory measurements, moreover, practical recommendations are made for future applications of the numerical solver.

Exergy and exergo-environmental analysis of a 660 MW supercritical coal-fired power plant

This paper presents the exergy and exergo-environmental analysis of the 660 MW supercritical coal-fired unit situated in western India. The study is based upon the SPECO approach, which is followed in the case of exergoeconomic analysis. The proposed research includes evaluating values of exergy of each stream, using F and P principal to find out exergy of fuel, the product of each component, and its exegetic efficiency. It further includes allocation of Eco-indicator 99 assessment grading values to perform life cycle assessment of each component, solving simultaneous equations by suitable solver formed from the environmental impact of components to calculate the exergo-environmental factor of each stream. The exergy efficiency obtained for the supercritical unit is 35.54%. The effect of variation of exergetic efficiency with an inlet pressure of high-pressure turbine for major components is studied. The major components such as steam generator, high-pressure turbine, intermediate-pressure turbine, low-pressure turbine, condensate extraction pump, and boiler feed pump exhibit higher exergetic efficiency at 247 bar. An increase in environmental impact in the exergy flows has been identified for components with having heat, work or fuel at the inlet. The cooling water used in condenser and flues gases exhausted from the supercritical power plant represent the environmental impact rate per unit electricity generated of 2.89 and 3.689 mPts kWh−1. The electricity generated had an environmental impact rate of 39.71 mPts kWh−1. The environmental impact per unit of electricity generated for 660 MW supercritical power plant is evaluated as 46.29 mPts kWh−1. The result obtained from semi-empirical model for environmental impact per unit of electricity is validated by comparing with subcritical and ultra-critical coal-fired power plant. The supercritical coal-fired power plant is proved to be better technology as compared with subcritical coal-fired power plant. The exergo-environmental variable for the steam generator offers high potential to reduce environmental impact by introducing an exhaust gas treatment unit.

Spectral Galerkin schemes for a class of multi-order fractional pantograph equations

In this paper, we study and present a spectral numerical technique for solving a general class of multi-order fractional pantograph equations with varying coefficients and systems of pantograph equations. In this study, the spectral Galerkin approach in combination with the properties of shifted Legendre polynomials is used to reduce such equations to systems of algebraic equations, which are solved using any suitable solver. As far as the authors know, this is the first attempt to deal with fractional pantograph equations via spectral Galerkin approach. The errors and convergence of the adopted approach are rigorously analyzed. The efficiency and accuracy of the technique are tested by considering five different examples, to ensure that the suggested approach is more accurate than the existing other techniques. The obtained results in this paper are comparing favorably with those published by other researchers and with the existing exact solutions, whenever possible.

TOTAL: Optimal Protection Strategy Against Perfect and Imperfect False Data Injection Attacks on Power Grid Cyber–Physical Systems

This article explores the problem of protection against false data injection attacks (FDIAs) on the power system state estimation. Although many research works have been reported previously to solve the same problem, yet most of them are only for perfect FDIAs. To address the problem reasonably, all related factors influencing the success probability and corresponding attack impact of imperfect FDIAs should also be considered. Based on such considerations, a topology, parameter, accuracy, level (TOTAL) protection strategy considering all corresponding factors is proposed. The TOTAL protection strategy minimizes the attack impact of typical imperfect FDIAs (single measurement attacks) while defending against typical perfect FDIAs (single-state variable attacks). Depending on whether the protection scheme contains phasor measurement units (PMUs), we formulate the meter selection as a linear binary programming or integer programming problem, which can be solved by suitable solvers. The proposed strategy is compared with the existing methods in the literature and evaluated using the standard IEEE test cases.

A numerical treatment of the two-dimensional multi-term time-fractional mixed sub-diffusion and diffusion-wave equation

In this paper, we consider an important kind of fractional partial differential equations, namely multi-term time-fractional mixed sub-diffusion and diffusion-wave equation. The crucial importance of the considered equation is due to the fact that it generalizes some substantial types of fractional differential equations that can be widely used in describing many real-life phenomena, some of these equations are the time-fractional sub-diffusion, time-fractional diffusion-wave and time-fractional diffusion equations. In this study, the 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation is transformed to its integrated form with respect to time. An extension of the operational matrix of second-order derivative to the 2D case is used in combination with the operational matrix of fractional-order integrals and the time-space spectral collocation method to reduce such equations to systems of algebraic equations, which are solved using any suitable solver. As far as the authors know, this is the first attempt to deal with 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation via a spectral approach. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. Our results confirm that nonlocal numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account.

A new interior point solver with generalized correntropy for multiple gross error suppression in state estimation

This paper provides an answer to the problem of State Estimation (SE) with multiple simultaneous gross errors, based on Generalized Error Correntropy instead of Least Squares and on an interior point method algorithm instead of the conventional Gauss–Newton algorithm. The paper describes the mathematical model behind the new SE cost function and the construction of a suitable solver and presents illustrative numerical cases. The performance of SE with the data set contaminated with up to five simultaneous gross errors is assessed with confusion matrices, identifying false and missed detections. The superiority of the new method over the classical Largest Normalized Residual Test is confirmed at a 99% confidence level in a battery of tests. Its ability to address cases where gross errors fall on critical measurements, critical sets or leverage points is also confirmed at the same level of confidence.

Optimized ellipse packings in regular polygons

We present model development and numerical solution approaches to the problem of packing a general set of ellipses without overlaps into an optimized polygon. Specifically, for a given set of ellipses, and a chosen integer m ≥ 3, we minimize the apothem of the regular m-polygon container. Our modeling and solution strategy is based on the concept of embedded Lagrange multipliers. To solve models with up to n ≤ 10 ellipses, we use the LGO solver suite for global–local nonlinear optimization. In order to reduce increasing runtimes, for model instances with 10 ≤ n ≤ 20 ellipses, we apply local search launching the Ipopt solver from selected random starting points. The numerical results demonstrate the applicability of our modeling and optimization approach to a broad class of highly non-convex ellipse packing problems, by consistently returning good quality feasible solutions in all (231) illustrative model instances considered here.