Research

19 pages, 2725 KiB

Open AccessArticle

An Innovative Numerical Method Utilizing Novel Cubic B-Spline Approximations to Solve Burgers’ Equation

by Ishtiaq Ali, Muhammad Yaseen, Muhammad Abdullah, Sana Khan and Fethi Bin Muhammad Belgacem

Mathematics 2023, 11(19), 4079; https://doi.org/10.3390/math11194079 - 26 Sep 2023

Cited by 1 | Viewed by 1350

Burgers’ equation is a nonlinear partial differential equation that appears in various areas of physics and engineering. Finding accurate and efficient numerical methods to solve this equation is crucial for understanding complex fluid flow phenomena. In this study, we propose a spline-based numerical [...] Read more.

Burgers’ equation is a nonlinear partial differential equation that appears in various areas of physics and engineering. Finding accurate and efficient numerical methods to solve this equation is crucial for understanding complex fluid flow phenomena. In this study, we propose a spline-based numerical technique for the numerical solution of Burgers’ equation. The space derivative is discretized using cubic B-splines with new approximations for the second order. Typical finite differences are used to estimate the time derivative. Additionally, the scheme undergoes a stability study to ensure minimal error accumulation, and its convergence is investigated. The primary advantage of this scheme is that it generates an approximate solution as a smooth piecewise continuous function, enabling approximation at any point within the domain. The scheme is subjected to a numerical study, and the obtained results are compared to those previously reported in the literature to demonstrate the effectiveness of the proposed approach. Overall, this study aims to contribute to the development of efficient and accurate numerical methods for solving Burgers’ equation. The spline-based approach presented herein has the potential to advance our understanding of complex fluid flow phenomena and facilitate more reliable predictions in a range of practical applications. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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14 pages, 348 KiB

Open AccessArticle

Numerical Solutions of Stochastic Differential Equations with Jumps and Measurable Drifts

by Maryam Siddiqui, Mhamed Eddahbi and Omar Kebiri

Mathematics 2023, 11(17), 3755; https://doi.org/10.3390/math11173755 - 31 Aug 2023

Cited by 1 | Viewed by 1055

Abstract

This paper deals with numerical analysis of solutions to stochastic differential equations with jumps (SDEJs) with measurable drifts that may have quadratic growth. The main tool used is the Zvonkin space transformation to eliminate the singular part of the drift. More precisely, the [...] Read more.

This paper deals with numerical analysis of solutions to stochastic differential equations with jumps (SDEJs) with measurable drifts that may have quadratic growth. The main tool used is the Zvonkin space transformation to eliminate the singular part of the drift. More precisely, the idea is to transform the original SDEJs to standard SDEJs without singularity by using a deterministic real-valued function that satisfies a second-order differential equation. The Euler–Maruyama scheme is used to approximate the solution to the equations. It is shown that the rate of convergence is

\frac{1}{2}

. Numerically, two different methods are used to approximate solutions for this class of SDEJs. The first method is the direct approximation of the original equation using the Euler–Maruyama scheme with specific tests for the evaluation of the singular part at simulated values of the solution. The second method consists of taking the inverse of the Euler–Maruyama approximation for Zvonkin’s transformed SDEJ, which is free of singular terms. Comparative analysis of the two numerical methods is carried out. Theoretical results are illustrated and proved by means of an example. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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20 pages, 7179 KiB

Open AccessArticle

An Improved Neural Particle Method for Complex Free Surface Flow Simulation Using Physics-Informed Neural Networks

by Kaixuan Shao, Yinghan Wu and Suizi Jia

Mathematics 2023, 11(8), 1805; https://doi.org/10.3390/math11081805 - 11 Apr 2023

Cited by 1 | Viewed by 1721

Abstract

The research on free surface flow is of great interest in fluid mechanics, with the primary task being the tracking and description of the motion of free surfaces. The development of numerical simulation techniques has led to the application of new methods in [...] Read more.

The research on free surface flow is of great interest in fluid mechanics, with the primary task being the tracking and description of the motion of free surfaces. The development of numerical simulation techniques has led to the application of new methods in the study of free surface flow problems. One such method is the Neural Particle Method (NPM), a meshless approach for solving incompressible free surface flow. This method is built on a Physics-Informed Neural Network (PINN), which allows for training and solving based solely on initial and boundary conditions. Although the NPM is effective in dealing with free surface flow problems, it faces challenges in simulating more complex scenarios due to the lack of additional surface recognition algorithms. In this paper, we propose an improved Neural Particle Method (INPM) to better simulate complex free surface flow. Our approach involves incorporating alpha-shape technology to track and recognize the fluid boundary, with boundary conditions updated constantly during operation. We demonstrate the effectiveness of our proposed method through three numerical examples with different boundary conditions. The result shows that: (1) the addition of a surface recognition module allows for the accurate tracking and recognition of the fluid boundary, enabling more precise imposition of boundary conditions in complex situations; (2) INPM can accurately identify the surface and calculate even when particles are unevenly distributed. Compared with traditional meshless methods, INPM offers a better solution for dealing with complex free surface flow problems that involve random particle distribution. Our proposed method can improve the accuracy and stability of numerical simulations for free surface flow problems. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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13 pages, 3583 KiB

Open AccessArticle

Integration of Fuzzy Ontologies and Neural Networks in the Detection of Time Series Anomalies

by Vadim Moshkin, Dmitry Kurilo and Nadezhda Yarushkina

Mathematics 2023, 11(5), 1204; https://doi.org/10.3390/math11051204 - 01 Mar 2023

Cited by 1 | Viewed by 1154

Abstract

This paper explores an approach to solving the problem of detecting time series anomalies, taking into account the specifics of the subject area. We propose a method based on the integration of a neural network with long short-term memory (LSTM) and Fuzzy OWL [...] Read more.

This paper explores an approach to solving the problem of detecting time series anomalies, taking into account the specifics of the subject area. We propose a method based on the integration of a neural network with long short-term memory (LSTM) and Fuzzy OWL (Fuzzy Web Ontology Language) ontology. A LSTM network is used for the mathematical search for anomalies in the first stage. The fuzzy ontology filters the detection results and draws an inference for decision making in the second stage. The ontology contains a formalized representation of objects in the subject area and inference rules that select only those anomaly values that correspond to this subject area. In the article, we propose the architecture of a software system that implements this approach. Computational experiments were carried out on free data of technical characteristics of drilling rigs. The experiments showed high efficiency, but not the maximum efficiency of the proposed approach. In the future, we plan to select a more efficient neural network architecture for mathematical anomaly detection. We also plan to develop an algorithm for automatically filling the rules of inference into the ontology when analyzing text sources. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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11 pages, 401 KiB

Open AccessArticle

New Development of Variational Iteration Method Using Quasilinearization Method for Solving Nonlinear Problems

by Vikash Kumar Sinha and Prashanth Maroju

Mathematics 2023, 11(4), 935; https://doi.org/10.3390/math11040935 - 12 Feb 2023

Cited by 4 | Viewed by 1213

Abstract

In this paper, we developed a new variational iteration method using the quasilinearization method and Adomian polynomial to solve nonlinear differential equations. The convergence analysis of our new method is also discussed under the Lipschitz continuity condition in Banach space. Some application problems [...] Read more.

In this paper, we developed a new variational iteration method using the quasilinearization method and Adomian polynomial to solve nonlinear differential equations. The convergence analysis of our new method is also discussed under the Lipschitz continuity condition in Banach space. Some application problems are included to test the efficacy of our proposed method. The behavior of the method is investigated for different values of parameter t. This is a powerful technique for solving a large number of nonlinear problems. Comparisons of our technique were made with the available exact solution and existing methods to examine the applicability and efficiency of our approach. The outcome revealed that the proposed method is easy to apply and converges to the solution very fast. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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18 pages, 626 KiB

Open AccessArticle

Adaptive Fuzzy Predictive Approach in Control

by Anton A. Romanov, Aleksey A. Filippov and Nadezhda G. Yarushkina

Mathematics 2023, 11(4), 875; https://doi.org/10.3390/math11040875 - 09 Feb 2023

Cited by 1 | Viewed by 891

Abstract

This article studies the approach to solving the problem of controlling the complex organizational and technical systems based on hybrid models. We propose a new component of intelligent decision support that is integrated with control systems. The proposed component is based on fuzzy [...] Read more.

This article studies the approach to solving the problem of controlling the complex organizational and technical systems based on hybrid models. We propose a new component of intelligent decision support that is integrated with control systems. The proposed component is based on fuzzy logic and knowledge engineering. We present a model of ontology to form the context of data analysis and time series modeling. The ontological context allows us to represent trends of the analyzed object indicators. An expert can add a set of fuzzy rules to the ontology for systems control based on the fuzzy inference. The proposed approach allows reducing the time of analysis and interpretation of the results. Experimental results confirm the correctness and effectiveness of the approach proposed in this article. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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15 pages, 527 KiB

Open AccessArticle

An Aggregation-Based Algebraic Multigrid Method with Deflation Techniques and Modified Generic Factored Approximate Sparse Inverses

by Anastasia A. Natsiou, George A. Gravvanis, Christos K. Filelis-Papadopoulos and Konstantinos M. Giannoutakis

Mathematics 2023, 11(3), 640; https://doi.org/10.3390/math11030640 - 27 Jan 2023

Cited by 1 | Viewed by 1046

Abstract

In this paper, we examine deflation-based algebraic multigrid methods for solving large systems of linear equations. Aggregation of the unknown terms is applied for coarsening, while deflation techniques are proposed for improving the rate of convergence. More specifically, the V-cycle strategy is adopted, [...] Read more.

In this paper, we examine deflation-based algebraic multigrid methods for solving large systems of linear equations. Aggregation of the unknown terms is applied for coarsening, while deflation techniques are proposed for improving the rate of convergence. More specifically, the V-cycle strategy is adopted, in which, at each iteration, the solution is computed by initially decomposing it utilizing two complementary subspaces. The approximate solution is formed by combining the solution obtained using multigrids and deflation. In order to improve performance and convergence behavior, the proposed scheme was coupled with the Modified Generic Factored Approximate Sparse Inverse preconditioner. Furthermore, a parallel version of the multigrid scheme is proposed for multicore parallel systems, improving the performance of the techniques. Finally, characteristic model problems are solved to demonstrate the applicability of the proposed schemes, while numerical results are given. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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24 pages, 643 KiB

Open AccessArticle

Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format

by Boian Alexandrov, Gianmarco Manzini, Erik W. Skau, Phan Minh Duc Truong and Radoslav G. Vuchov

Mathematics 2023, 11(3), 534; https://doi.org/10.3390/math11030534 - 19 Jan 2023

Cited by 1 | Viewed by 1807

Abstract

Numerical integration is a basic step in the implementation of more complex numerical algorithms suitable, for example, to solve ordinary and partial differential equations. The straightforward extension of a one-dimensional integration rule to a multidimensional grid by the tensor product of the spatial [...] Read more.

Numerical integration is a basic step in the implementation of more complex numerical algorithms suitable, for example, to solve ordinary and partial differential equations. The straightforward extension of a one-dimensional integration rule to a multidimensional grid by the tensor product of the spatial directions is deemed to be practically infeasible beyond a relatively small number of dimensions, e.g., three or four. In fact, the computational burden in terms of storage and floating point operations scales exponentially with the number of dimensions. This phenomenon is known as the curse of dimensionality and motivated the development of alternative methods such as the Monte Carlo method. The tensor product approach can be very effective for high-dimensional numerical integration if we can resort to an accurate low-rank tensor-train representation of the integrand function. In this work, we discuss this approach and present numerical evidence showing that it is very competitive with the Monte Carlo method in terms of accuracy and computational costs up to several hundredths of dimensions if the integrand function is regular enough and a sufficiently accurate low-rank approximation is available. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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14 pages, 3206 KiB

Open AccessArticle

Adaptive Load Incremental Step in Large Increment Method for Elastoplastic Problems

by Baorang Cui, Jingxiu Zhang and Yong Ma

Mathematics 2023, 11(3), 524; https://doi.org/10.3390/math11030524 - 18 Jan 2023

Viewed by 1389

Abstract

As a force-based finite element method (FEM), large increment method (LIM) shows unique advantages in material nonlinearity problems. In LIM for material nonlinearity analysis, adaptive load incremental step is a fundamental step for its successful application. In this work, a strategy to automatically [...] Read more.

As a force-based finite element method (FEM), large increment method (LIM) shows unique advantages in material nonlinearity problems. In LIM for material nonlinearity analysis, adaptive load incremental step is a fundamental step for its successful application. In this work, a strategy to automatically refine the load incremental step is proposed in the framework of LIM. The adaptive load incremental step is an iterative process based on the whole loading process, and the location and number of post-refined incremental steps are determined by the posteriori error of energy on the pre-refined incremental steps. Furthermore, the iterative results from the pre-refined incremental steps can be utilized as the initial value to calculate the result for the post-refined incremental steps, which would significantly improve the computational accuracy and efficiency. The strategy is demonstrated using a two-dimensional example with a bilinear hardening material model under cyclic loading, which verifies the accuracy and efficiency of the strategy in LIM. Compared with the displacement-based FEM, which relies upon a step-by-step incremental approach stemming from flow theory, the adaptive load incremental step based on the whole loading process of LIM can avoid the cumulative errors caused by step-by-step in global stage and can quantify the accuracy of results. This work provides a guidance for the practical application of LIM in nonlinear problems. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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18 pages, 11411 KiB

Open AccessArticle

Fracture Process and Failure Mode of Brazilian Discs with Cracks of Different Angles: A Numerical Study

by Xiaoyan Luo, Guoyan Zhao, Peng Xiao and Wengang Zhao

Mathematics 2022, 10(24), 4808; https://doi.org/10.3390/math10244808 - 17 Dec 2022

Viewed by 1171

Abstract

In order to determine the effect of internal cracks on the tensile failure of materials, a hybrid finite–discrete element method was used to analyze the Brazilian disc test with cracks of different angles. When the pre-crack angle is between 0° and 60°, the [...] Read more.

In order to determine the effect of internal cracks on the tensile failure of materials, a hybrid finite–discrete element method was used to analyze the Brazilian disc test with cracks of different angles. When the pre-crack angle is between 0° and 60°, the wing crack is initiated from the pre-crack end. When the pre-crack is 90°, the crack initiated from the pre-crack center. When the pre-crack angle is between 0° and 60°, the maximum principal stress and plastic strain are concentrated at the pre-crack end. When the pre-crack angle is 90°, the maximum principal stress and plastic strain are concentrated in the pre-crack center. As the crack angle increased from 0° to 90°, the failure mode of Brazilian discs with cracks transits from splitting into two parts to splitting into four parts. The influence of crack length is further studied. When the crack length is less than 5 mm, the crack angle has little influence on the disc failure mode; Brazilian discs with cracks of different angles undergoes splitting failure along the loading axis. When the crack length is larger than 5 mm, the crack angle has a great effect on the disc failure mode. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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16 pages, 5636 KiB

Open AccessArticle

Numerical Solving Method for Jiles-Atherton Model and Influence Analysis of the Initial Magnetic Field on Hysteresis

by Guangming Xue, Hongbai Bai, Tuo Li, Zhiying Ren, Xingxing Liu and Chunhong Lu

Mathematics 2022, 10(23), 4431; https://doi.org/10.3390/math10234431 - 24 Nov 2022

Cited by 2 | Viewed by 2032

Abstract

The Jiles-Atherton model was widely used in the description of the system with hysteresis, and the solution for the model was important for real-time and high-precision control. The secant method was used for solving anhysteretic magnetization and its initial values were optimized for [...] Read more.

The Jiles-Atherton model was widely used in the description of the system with hysteresis, and the solution for the model was important for real-time and high-precision control. The secant method was used for solving anhysteretic magnetization and its initial values were optimized for faster convergence. Then, the Fourth Order Runge-Kutta method was employed to solve magnetization and the required computation cycles were supplied for stable results. Based on the solving method, the effect of the nonzero initial magnetic field on the magnetization was discussed, including the commonly used linear model of the square of magnetization under the medium initial value. From computations, the proposed secant iteration method, with supplied optimal initial values, greatly reduced the iterative steps compared to the fixed-point iteration. Combined with the Fourth Order Runge-Kutta method under more than three cycles of calculations, stable hysteresis results with controllable precisions were acquired. Adjusting the initial magnetic field changed the result of the magnetization, which was helpless to promote the amplitude or improve the symmetry of magnetization. Furthermore, the linear model of the square of magnetization was unacceptable for huge computational errors. The proposed numerical solving method can supply fast and high-precision solutions for the Jiles-Atherton model and provide a basis for the application scope of typical linear assumption. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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22 pages, 8589 KiB

Open AccessArticle

Numerical Simulation Research of Bubble Characteristics and Bubble Departure Diameter in Subcooled Flow Boiling

by Jinfeng Wang, Bingjun Wang, Jing Xie, Ke Lei, Bo Yu and Yuhang Sun

Mathematics 2022, 10(21), 4103; https://doi.org/10.3390/math10214103 - 03 Nov 2022

Cited by 3 | Viewed by 1147

Abstract

Three-dimensional subcooled flow boiling of R134a in a horizontal tube was simulated by a VOF (volume of fluid) model combined with the level set method. Bubble characteristics were explored at heat flux of 0.3 MW/m², inlet subcooling of 3 K, and [...] Read more.

Three-dimensional subcooled flow boiling of R134a in a horizontal tube was simulated by a VOF (volume of fluid) model combined with the level set method. Bubble characteristics were explored at heat flux of 0.3 MW/m², inlet subcooling of 3 K, and inlet velocity of 0.4 m/s. It was observed that five representative bubbles occurred in subcooled flow boiling, including sliding bubble, coalescing bubble, non-departed bubble, bouncing bubble, and continuous-boiling bubble. The results showed that the bubble radial velocity was an important factor of bubble departure after a sliding process. Moreover, the effect of heat flux, inlet velocity, and inlet subcooling on bubble departure diameter were investigated. The departure diameter increased with increasing inlet velocity from 0.2 to 0.4 m/s and heat flux from 0.2 to 0.4 MW/m², while diameter decreased with inlet subcooling from 3 to 10 K. Finally, based on the influence of heat flux, inlet velocity, and inlet subcooling on average departure diameter of the bubble except the coalescing bubble, a model was proposed to predict the average departure diameter. The deviation of the model was within 5%. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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9 pages, 347 KiB

Open AccessArticle

Exponentially-Fitted Fourth-Derivative Single-Step Obrechkoff Method for Oscillatory/Periodic Problems

by Ashiribo Senapon Wusu, Olusola Aanu Olabanjo and Manuel Mazzara

Mathematics 2022, 10(14), 2392; https://doi.org/10.3390/math10142392 - 07 Jul 2022

Cited by 1 | Viewed by 1142

Abstract

The quest for accurate and more efficient methods for solving periodic/oscillatory problems is gaining more attention in recent time. This paper presents the construction and implementation of a family of exponentially-fitted Obrechkoff methods using a six-step flowchart discussed in the literature. A single-step [...] Read more.

The quest for accurate and more efficient methods for solving periodic/oscillatory problems is gaining more attention in recent time. This paper presents the construction and implementation of a family of exponentially-fitted Obrechkoff methods using a six-step flowchart discussed in the literature. A single-step Obrechkoff method involving terms up to the fourth derivative was used as the base method. We also present the stability and convergence properties of the constructed family of methods. Two numerical examples were used to illustrate the performance of the constructed methods. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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24 pages, 3595 KiB

Open AccessFeature PaperArticle

Applying the Random Forest Method to Improve Burner Efficiency

by Vladislav Kovalnogov, Ruslan Fedorov, Vladimir Klyachkin, Dmitry Generalov, Yulia Kuvayskova and Sergey Busygin

Mathematics 2022, 10(12), 2143; https://doi.org/10.3390/math10122143 - 20 Jun 2022

Cited by 8 | Viewed by 2093

Abstract

Fuel power plants are one of the main sources of pollutant emissions, so special attention should be paid to improving the efficiency of the fuel combustion process. The mathematical modeling of processes in the combustion chamber makes it possible to reliably predict and [...] Read more.

Fuel power plants are one of the main sources of pollutant emissions, so special attention should be paid to improving the efficiency of the fuel combustion process. The mathematical modeling of processes in the combustion chamber makes it possible to reliably predict and find the best dynamic characteristics of the operation of a power plant, in order to quantify the emission of harmful substances, as well as the environmental and technical and economic efficiency of various regime control actions and measures, and the use of new types of composite fuels. The main purpose of this article is to illustrate how machine learning methods can play an important role in modeling and predicting the performance and control of the combustion process. The paper proposes a mathematical model of an unsteady turbulent combustion process, presents a model of a combustion chamber with a combined burner, and performs a numerical study using the STAR-CCM+ multidisciplinary platform. The influence of various input indicators on the efficiency of burner devices, which is evaluated by several parameters at the output, is investigated. In this case, three possible states of the burners are assumed: optimal, satisfactory and unsatisfactory. Full article

(This article belongs to the Special Issue Advanced Numerical Analysis and Scientific Computing)

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