Accuracy Of Numerical Calculations Research Articles

Enhanced shifted Jacobi operational matrices of integrals: spectral algorithm for solving some types of ordinary and fractional differential equations

We provide here a novel approach for solving IVPs in ODEs and MTFDEs numerically by means of a class of MSJPs. Using the SCM, we build OMs for RIs and RLFI for MSJPs as part of our process. These architectures guarantee accurate and efficient numerical computations. We provide theoretical assurances for the efficacy of an algorithm by establishing its convergence and error analysis features. We offer five numerical examples to prove that our method is accurate and applicable. Through these examples, we demonstrate the greater accuracy and efficiency of our approach by comparing our results with previously published findings. Tables and graphs show that the method produces exact and approximate solutions that agree quite well with each other.

Numerical analysis and experimental study of membrane-type acoustic metamaterial plate with X-shaped pendulum arm and cylindrical mass blocks

Membrane-type acoustic metamaterials have a favorable noise suppression effect. Hence, a membrane-type acoustic metamaterial plate (MAML) with an X-shaped pendulum arm and cylindrical mass blocks (CMB) was proposed in this paper. The theoretical model based on spring–mass systems and numerical simulation models of the membrane-type acoustic metamaterial cell (MAMC) were established under fixed and periodic boundary conditions to reveal the noise attenuation mechanism quantitatively, and the necessity of the CMB is discussed. Based on the results, it can be observed that the normal displacement of the membrane is nearly zero when the noise frequency is at the peak of the sound transmission loss (STL) curve. However, when the frequency is at the valley of the STL curve, the displacement is nonzero and fluctuates significantly. Meanwhile, by comparing the STL curves with and without the CMB, it was found that the MAMC performance is improved effectively by the CMB at low frequencies. The effective mass density of MAMC was found to be negative. To verify the accuracy of numerical calculations, an impedance tube experiment was conducted. Finally, orthogonal experiments were designed to describe the effects of the structural parameters a, l, and t on the effective bandwidth [Formula: see text], peak frequency [Formula: see text], and comprehensive index [Formula: see text] and to obtain the optimal structural parameter combinations for different indexes. This work further contributes to applying and developing the membrane-type acoustic metamaterial.

Modelling permanent magnet excited uniform fields with rational approximations

PurposeA novel method for modelling permanent magnets is investigated based on numerical approximations with rational functions. This study aims to introduce the AAA algorithm and other recently developed, cutting-edge mathematical tools, which provide outstandingly fast and accurate numerical computation of potentials and vector fields.Design/methodology/approachFirst, the AAA algorithm is briefly introduced along with its main variants and other advanced mathematical tools involved in the modelling. Then, the analysis of a circular Halbach array with a one-pole pair is carried out by means of the AAA-least squares method, focusing on vector potential and flux density in the bore and validating results by means of classic finite element software. Finally, the investigation is completed by a finite difference analysis.FindingsAAA methods for field analysis prove to be strikingly fast and accurate. Results are in excellent agreement with those provided by the finite element model, and the very good agreement with those from finite differences suggests future improvements. They are also easy programming; the MATLAB code is less than 200 lines. This indicates they can provide an effective tool for rapid analysis.Research limitations/implicationsAAA methods in magnetostatics are novel, but their extension to analogous physical problems seems straightforward. Being a meshless method, it is unlikely that local non-linearities can be considered. An aspect of particular interest, left for future research, is the capability of handling inhomogeneous domains, i.e. solving general interface problems.Originality/valueThe authors use cutting-edge mathematical tools for the modelling of complex physical objects in magnetostatics.

New Generalized Jacobi Polynomial Galerkin Operational Matrices of Derivatives: An Algorithm for Solving Boundary Value Problems

In this study, we present a novel approach for the numerical solution of high-order ODEs and MTVOFDEs with BCs. Our method leverages a class of GSJPs that possess the crucial property of satisfying the given BCs. By establishing OMs for both the ODs and VOFDs of the GSJPs, we integrate them into the SCM, enabling efficient and accurate numerical computations. An error analysis and convergence study are conducted to validate the efficacy of the proposed algorithm. We demonstrate the applicability and accuracy of our method through eight numerical examples. Comparative analyses with prior research highlight the improved accuracy and efficiency achieved by our approach. The recommended approach exhibits excellent agreement between approximate and precise results in tables and graphs, demonstrating its high accuracy. This research contributes to the advancement of numerical methods for ODEs and MTVOFDEs with BCs, providing a reliable and efficient tool for solving complex BVPs with exceptional accuracy.

Effect of head on the flow state of water in tailwater channel of the hydropower station

Hydropower station tailwater channel is often prone to elevated water levels and insufficient energy consumption, which leads to the influence of the output of the unit. To study the intrinsic connection between the water flow state and elevated water level in the tailwater channel, this paper investigates the water flow characteristics inside the tailwater channel of the Mupo Hydropower Station under different heads and verifies the accuracy of numerical calculations by constructing a physical model test platform. The results show that the maximum velocity at the bottom of the fluid domain near the anti-slope section of the tailwater channel decreases with the increase of head, while the velocity at the top of the fluid domain near the anti-slope section of the tailwater channel does not change much with the increase of head; a large vortex is formed in the middle of the upper part of the fluid domain in the anti-slope section of the tailwater channel; and the lower the head, the more obvious the vortex is. The results provide guidance for the design of tailwater channels at hydropower stations.

Evaluation Method and Analysis on Performance of Diffuser in Heat Storage Tank

The diffuser is a critical component in a heat storage tank, and its structure has an important influence on the thermal performance of the heat storage tank. At present, research on the structure of diffusers often focuses on the change of one single parameter, which results in the need for a comprehensive structure analysis of diffusers in heat storage tanks. This paper comprehensively considers the inlet diameter, hole distance, and hole diameter of the diffuser and the inlet conditions of the heat storage tank. Then, a new evaluation index, namely the non-uniformity coefficient of the diffuser, is proposed. The experiments verify the accuracy of numerical calculation, and the related empirical formula is summarized finally. The results show that the diffuser with a small non-uniformity coefficient can achieve thin temperature stratification and higher exergy efficiency. In other words, the non-uniformity coefficient is in good agreement with temperature stratification and the exergy efficiency of the heat storage tank. When the flow rate of the inlet device is 0.5 m/s, the exergy efficiency increases by nearly 30 percentage points and the non-uniformity coefficient of the No. 4 diffuser is 3.39%. In contrast, that of the No. 8 diffuser is 75.17%. This evaluation index is suitable for different diffuser types with high accuracy. It provides the theoretical and experimental basis for the structural design and selection of diffusers for heat storage tanks.

Study on the effects of season, geographical location, and altitude on thermal characteristics of large airship based on meteorological data

ABSTRACT Our team built models to accurately simulate the heat transfer inside and outside the airship. In this paper, we used the models to analyze the thermal characteristics of large airship at different seasons in five different regions and at different low altitudes using Fluent software. The results show that the seasons, geographical locations, and altitudes produce great effects on the thermal characteristics of airship by affecting the solar irradiation on the envelope surface, convection heat transfer, atmospheric parameters, etc. In order to validate the accuracy of numerical calculation, we built a testing platform and conducted the experiments. The measured data are in good agreement with the numerical results.

Enhanced shifted Jacobi operational matrices of derivatives: spectral algorithm for solving multiterm variable-order fractional differential equations

This paper presents a new way to solve numerically multiterm variable-order fractional differential equations (MTVOFDEs) with initial conditions by using a class of modified shifted Jacobi polynomials (MSJPs). As their defining feature, MSJPs satisfy the given initial conditions. A key aspect of our methodology involves the construction of operational matrices (OMs) for ordinary derivatives (ODs) and variable-order fractional derivatives (VOFDs) of MSJPs and the application of the spectral collocation method (SCM). These constructions enable efficient and accurate numerical computation. We establish the error analysis and the convergence of the proposed algorithm, providing theoretical guarantees for its effectiveness. To demonstrate the applicability and accuracy of our method, we present five numerical examples. Through these examples, we compare the results obtained with other published results, confirming the superiority of our method in terms of accuracy and efficiency. The suggested algorithm yields very accurate agreement between the approximate and exact solutions, which are shown in tables and graphs.

The s-version finite element method for non-linear material problems

This study intended to extend the s-version of the finite element method (s-version FEM) to cope with elastic–plastic problems. Compared with the conventional FEM, the s-version FEM, which overlays a set of local mesh with fine element size representing irregular features over the conventional FE mesh with coarse element size, can considerably simplify issues in domain discretisation with fewer degrees of freedom and provide accurate numerical predictions. However, most applications of the s-version FEM were limited to elastic problems only, leaving its applications to elastic–plastic problems almost blank owing to insufficient instructions on stress update when plasticity sets in. To address these issues, detailed instructions and formulations to cope with plasticity problems with the s-version FEM were presented for its first time. The recursive element subdivision technique was implemented to generalise its application where local elements intersect with global element edges. A 3D stress concentration problem with linear/non-linear material properties was analysed; their numerical results were compared with the exact solutions and those obtained from the conventional FEM with very fine elements. The comparison highlights the flexibility of the s-version FEM in domain discretisation and its superior accuracy in numerical calculations, thereby exhibiting high potential applications in structural integrity assessment with complex structures, geometric defects, and material non-linearity.

Continuation of global solution curves using global parameters

This paper provides both the theoretical results and numerical calculations of global solution curves, by continuation in global parameters. Each point on the solution curves is computed directly as the global parameter is varied, so that all of the turns that the solution curves make, as well as their different branches, appear automatically on the computer screen. For radial p-Laplace equations we present a simplified derivation of the regularizing transformation from P. Korman (2015), and use this transformation for more accurate numerical computations. While for p>2 the solutions are not of class C2, we show that they are of the form w(rp2(p−1)), where w(z) is of class C2. Bifurcation diagrams are also calculated for non-autonomous problems, and for the fourth order equations modeling elastic beams. We show that the first harmonic of the solution can also serve as a global parameter.

Optimization of geometric parameters of hydraulic turbine runner in turbine mode based on ISMA and BPNN

AbstractThe hydraulic turbine in turbine mode (TMHT) has significant advantages in residual energy recovery, but rapid optimization of its runner parameters has always been a challenge. To address this issue, the optimization algorithm ISMA‐BPNN is proposed based on the improved slime mold algorithm (ISMA) and back propagation neural network (BPNN). First, an efficiency characteristic neural network (ECNN) and a water head characteristic neural network (HCNN), which take the geometric parameters of the runner as input, and the efficiency and water head as outputs, respectively, are constructed by combining the orthogonal test‐based sample data, computational fluid dynamics (CFD), and BPNN. Then, the slime mold algorithm is improved and the optimal runner geometric parameters are obtained based on the ISMA, ECNN, and HCNN, to achieve rapid optimization of the TMHT runner. Finally, the CFD‐based numerical calculation accuracy is verified through real machine tests, and the feasibility of the ISMA‐BPNN‐based rapid optimization strategy for TMHT performance is further verified through CFD numerical simulation.

The effect of tip clearance on the rotor-stator interaction and noise of marine 1.5-stage compressor

Tip-clearance (TC) noise is significant in compressors and has garnered scholarly attention for the isolated rotor. However, applying these findings to marine compressors with rotor-stator-interaction (RSI) is challenging. In this paper, a comprehensive analysis was given of a marine 1.5-stage compressor. The investigation included five different TC configurations (0, 0.25, 0.5, 0.75, and 1 mm) and the RSI was a focal point of the study. The accuracy of numerical calculations of the flow field was verified using experimental data. The results reveal a new phenomenon; the radial separation flow of the rotor’s trailing edge is mixed with the TC flow and a wake above the 90 span of the blade is formed. An understanding of this mixing mechanism is crucial for model that describes the effect of TC on the RSI. The single-tone sound source of the stator’s leading-edge tip area is the smallest when the TC is 0.25 mm. In contrast, the single-tone sound source is largest when the TC is equal to zero. Investigation of different TCs in three acoustic environments shows no obvious disparity in the single-tone acoustic power response of various TCs. There is, however, a correlation between the flow mixing mechanism and the noise.

Transient eddy current analysis for a spiral gradient pulse

Eddy currents are induced in the metallic structures of MRI machines due to the rapid switching of gradient fields generated by gradient coils. Several undesirable effects are associated with the induced eddy currents such as heat, acoustic noise, and MR image distortions. Accurate transient eddy currents numerical computations are required to predict and ameliorate such effects. Spiral gradient waveforms are of importance, particularly for fast MRI acquisition applications. For mathematical convenience, previously published work is mostly concerned with transient eddy currents computations associated with trapezoidal gradient waveforms; where spiral gradient waveforms were not considered. We recently presented preliminary transient eddy currents computations induced by an amplitude-modulated sinusoidal pulse in the scanner’s cryostat. In this work, we present a full computational framework for transient eddy currents induced by a spiral gradient waveform. A mathematical model for transient eddy currents involving the spiral pulse was derived and presented in detail using the circuit equation. Computations were implemented using a tailored multilayer integral method (TMIM) and results were compared to Ansys eddy currents analysis for cross-validation. The transient response of resultant fields generated by both an unshielded transverse coil driven by a spiral waveform was computed showing high agreement between Ansys and TMIM; albeit with high computational efficiency concerning time and memory. For further validation, computations for a shielded transverse coil were performed showing how eddy currents effects are reduced.

The simple shifted fracture method

AbstractWe propose a simplified version of the Shifted Fracture Method (SFM), which relies on an approximate fracture geometry representation combined with approximate interface conditions. In particular, we show that the complexities of earlier SFM implementations, associated with the use of a mixed variational formulation of the equilibrium equations, can be avoided. This approach has minimal impact on the overall accuracy of numerical computations but leads to considerable gains in performance. These outcomes can be justified because the proposed simplified SFM accurately accounts for the area of the fracture surface and prevents the severe mesh dependencies observed in classical cohesive zone models or node‐release/element‐deletion techniques. We also show that the proposed approach can be combined with a phase‐field model to more easily capture branching and merging of fractures.

Higher corrections of the Ilkovich equation

• Ilkovic polarography equation derived and augmented. • Asymptotic analysis leads systematically to higher-order terms. • Newman’s 1st-order correction verified; 2nd-order correction computed for the first time. • Asymptotic methods compared to accurate numerical computation. A short-time asymptotic analysis is performed to establish corrections of the Ilkovich equation, which describes the polarographic response of a dropping mercury electrode. The convective diffusion equation governing diffusion limited reactant flux for small drop times is solved by a regular perturbation based on powers of the sixth root of time. This produces a framework within which higher terms of the Ilkovich equation can be derived systematically. As well as reproducing Ilkovich’s original formula and verifying Newman’s correction of Koutecky’s first-order term, we calculate the second-order term for the first time. The calculation is compared to the Newman–Levich procedure and tested against numerical simulations with finite-element software.

Quantitative two/three-dimensional spatial characterization and fluid transport prediction of macro/micropores in Gaomiaozi bentonite

The sealing performance of a bentonite barrier is highly dependent on its seepage characteristics, which are directly related to the characteristics of its pore structure. Based on scanning electron microscopy (SEM) and focused ion beam-SEM (FIB-SEM), the pore structure of bentonite was characterized at different scales. First, a reasonable gray threshold was determined through back analysis, and the image was binarized based on the threshold. In addition, binary images were used to analyze bentonite's pore structure (porosity and pore size distribution). Furthermore, the effects of different algorithms on the pore structure characterization were evaluated. Then, permeability calculations were performed based on the previous pore structure characteristics and a modified permeability prediction model. For permeability prediction based on the three-dimensional model, the effect of pore tortuosity was also considered. Finally, the accuracy of numerical calculations was verified by conducting macroscopic gas and alcohol permeability experiments. This approach provides a better understanding of the microscale mechanism of gas transport in bentonite and the importance of pore structures at different scales in determining its seepage characteristics. • The influence of image magnification on the quantitative characterization of the pore structure and the prediction of permeability is discussed. • The influence of different algorithms on the characterization of pore size distribution is discussed. • An improved permeability prediction model that can consider the effect of different pore sizes on seepage characteristics is proposed.

A whole high-accuracy numerical calculation system for the 1D Poisson equation by the interpolation finite difference method

The interpolation finite difference method (IFDM) allows free numerical analysis of elliptic partial differential equations over arbitrary domains. Conventionally, in the finite difference method (FDM), the calculation is performed using the second-order accuracy central difference. For engineering problems, second-order accuracy calculations are often sufficient. On the other hand, much research has been carried out to improve the accuracy of numerical calculations. Although there is much research in the FDM field, the development of numerical calculations by the spectral method is decisive in improving the calculation accuracy. Numerical calculations are usually performed by double precision calculations. If double precision calculations ensure 15 significant digits in floating point computing, such numerical calculations will be the ultimate goal to reach. A numerical calculation that does not seem to have an error even though it originally has an error is defined as a virtual error-zero (VE0) calculation. In this paper, we will examine an overall picture of high-accuracy numerical calculation by the IFDM in the numerical calculation of the 1D Poisson equation. It becomes clear that a VE0 calculation is always possible in the numerical calculation method, defined as the compact interpolation finite difference scheme [(m)].

Sufficient and Necessary Condition for the Asymptotic Stability of Stochastic Systems With Discrete Time Feedbacks and Applications

This article is devoted to reveal some essential features of stochastic control systems with sampled data (SCSwSD). First, it is shown by a proposition that the moment asymptotic stability of the underlying system is equivalent to that of any regular accurate numerical scheme under simple conditions, which is convenient to be structured specially for SCSwSD. This kind of principle provides a way for inferring moment asymptotic stability of SCSwSD by numerical simulations logically. As the first application of the proposition, the accurate scheme construction procedure is introduced in a general framework and illustrated for the quasi-linear models, the mean square asymptotic stability of linear SCSwSD is investigated. The restriction to the upper bound of the sampling period is confirmed by the way. As the second application of the proposition, the almost sure stability of a kind of controlled system with sampled noise is analyzed via the discrete scheme approach. The concepts of accurate numerical computation and simulation (ANCS) are proposed. A distinctive character, sampled data based control (SDBC) only, for SDBC is reported and studied preliminarily based on ANCS and the equivalence proposition. Some important remarks are given as further analyses on related issues. Finally, numerical examples are given to illustrate the applications of the theoretical results.

Influence of torsion on icing process of transmission lines

Icing threatens the safe and stable operation of transmission lines. Accurate prediction of the icing process of conductor can help the work of anti-icing and disaster reduction of transmission lines. Generally, the icing of conductor is accompanied by the torsion process of conductor. Because of the interaction of conductor torsion and icing, there is a big difference of the icing shape and rate between different positions on a transmission line. In order to improve the accuracy of numerical calculation of conductor icing, a dynamic numerical calculation model of conductor icing torsion is established based on hydrodynamics and basic mechanics in this paper. Comprehensively, the model takes the effects of torsion angle and torsion icing shape on the icing process into consideration, including the effects on airflow filed and trajectories of water droplets in the air. Through the simulation, the change law of the torsion angle at different positions of the conductor, the influence characteristics of the conductor torsion on the icing process, and the differences of the icing torsion of conductor under different environmental parameters are analysed. The results show that, compared with the condition without torsion, the collision range of water droplets on the conductor is larger under the torsion condition, the average value of collision efficiency β1 is higher, and the icing rate is faster. The larger the wind velocity and the median volume diameter of water droplets in the air, the faster the conductor icing torsion development.

Singularity formation in the deterministic and stochastic fractional Burgers equation

This study is motivated by the question of how singularity formation and other forms of extreme behavior in nonlinear dissipative partial differential equations are affected by stochastic excitations. To address this question we consider the 1D fractional Burgers equation with additive colored noise as a model problem. This system is interesting, because in the deterministic setting it exhibits finite-time blow-up or a globally well-posed behavior depending on the value of the fractional dissipation exponent. The problem is studied by performing a series of accurate numerical computations combining spectrally-accurate spatial discretization with a Monte-Carlo approach. First, we carefully document the singularity formation in the deterministic system in the supercritical regime where the blow-up time is shown to be a decreasing function of the fractional dissipation exponent. Our main result for the stochastic problem is that there is no evidence for additive noise to regularize the evolution by suppressing singularities in the supercritical regime, or for the noise to trigger singularities in the subcritical regime. However, as the noise amplitude becomes large, the maximum existence time of the solution in the supercritical regime is shown to exhibit an increasingly non-Gaussian behavior. Analogous observations are also made for the maximum attained values of the enstrophy and the times when the maxima occur in the subcritical regime.