Mathematics

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25 pages, 923 KiB

Open AccessArticle

Real-Time Data Assimilation in Welding Operations Using Thermal Imaging and Accelerated High-Fidelity Digital Twinning

by Pablo Pereira Álvarez, Pierre Kerfriden, David Ryckelynck and Vincent Robin

Mathematics 2021, 9(18), 2263; https://doi.org/10.3390/math9182263 - 15 Sep 2021

Cited by 5 | Viewed by 1858

Welding operations may be subjected to different types of defects when the process is not properly controlled and most defect detection is done a posteriori. The mechanical variables that are at the origin of these imperfections are often not observable in situ. We [...] Read more.

Welding operations may be subjected to different types of defects when the process is not properly controlled and most defect detection is done a posteriori. The mechanical variables that are at the origin of these imperfections are often not observable in situ. We propose an offline/online data assimilation approach that allows for joint parameter and state estimations based on local probabilistic surrogate models and thermal imaging in real-time. Offline, the surrogate models are built from a high-fidelity thermomechanical Finite Element parametric study of the weld. The online estimations are obtained by conditioning the local models by the observed temperature and known operational parameters, thus fusing high-fidelity simulation data and experimental measurements. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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25 pages, 3222 KiB

Open AccessArticle

Node Generation for RBF-FD Methods by QR Factorization

by Tony Liu and Rodrigo B. Platte

Mathematics 2021, 9(16), 1845; https://doi.org/10.3390/math9161845 - 05 Aug 2021

Cited by 2 | Viewed by 1676

Abstract

Polyharmonic spline (PHS) radial basis functions (RBFs) have been used in conjunction with polynomials to create RBF finite-difference (RBF-FD) methods. In 2D, these methods are usually implemented with Cartesian nodes, hexagonal nodes, or most commonly, quasi-uniformly distributed nodes generated through fast algorithms. We [...] Read more.

Polyharmonic spline (PHS) radial basis functions (RBFs) have been used in conjunction with polynomials to create RBF finite-difference (RBF-FD) methods. In 2D, these methods are usually implemented with Cartesian nodes, hexagonal nodes, or most commonly, quasi-uniformly distributed nodes generated through fast algorithms. We explore novel strategies for computing the placement of sampling points for RBF-FD methods in both 1D and 2D while investigating the benefits of using these points. The optimality of sampling points is determined by a novel piecewise-defined Lebesgue constant. Points are then sampled by modifying a simple, robust, column-pivoting QR algorithm previously implemented to find sets of near-optimal sampling points for polynomial approximation. Using the newly computed sampling points for these methods preserves accuracy while reducing computational costs by mitigating stencil size restrictions for RBF-FD methods. The novel algorithm can also be used to select boundary points to be used in conjunction with fast algorithms that provide quasi-uniformly distributed nodes. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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13 pages, 4148 KiB

Open AccessArticle

Nanofluid Flow on a Shrinking Cylinder with Al₂O₃ Nanoparticles

by Iskandar Waini, Anuar Ishak and Ioan Pop

Mathematics 2021, 9(14), 1612; https://doi.org/10.3390/math9141612 - 08 Jul 2021

Cited by 18 | Viewed by 2291

Abstract

This study investigates the nanofluid flow towards a shrinking cylinder consisting of Al₂O₃ nanoparticles. Here, the flow is subjected to prescribed surface heat flux. The similarity variables are employed to gain the similarity equations. These equations are solved via the [...] Read more.

This study investigates the nanofluid flow towards a shrinking cylinder consisting of Al₂O₃ nanoparticles. Here, the flow is subjected to prescribed surface heat flux. The similarity variables are employed to gain the similarity equations. These equations are solved via the bvp4c solver. From the findings, a unique solution is found for the shrinking strength

λ \geq - 1

. Meanwhile, the dual solutions are observed when

λ_{c} < λ < - 1

. Furthermore, the friction factor

R e_{x}^{1 / 2} C_{f}

and the heat transfer rate

R e_{x}^{- 1 / 2} N u_{x}

increase with the rise of Al₂O₃ nanoparticles

φ

and the curvature parameter

γ

. Quantitatively, the rates of heat transfer

R e_{x}^{- 1 / 2} N u_{x}

increase up to 3.87% when

φ

increases from 0 to 0.04, and 6.69% when

γ

increases from 0.05 to 0.2. Besides, the profiles of the temperature

θ (η)

and the velocity

f ’ (η)

on the first solution incline for larger

γ

, but their second solutions decline. Moreover, it is noticed that the streamlines are separated into two regions. Finally, it is found that the first solution is stable over time. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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14 pages, 7045 KiB

Open AccessArticle

Application of Artificial Intelligence and Gamma Attenuation Techniques for Predicting Gas–Oil–Water Volume Fraction in Annular Regime of Three-Phase Flow Independent of Oil Pipeline’s Scale Layer

by Abdulaziz S. Alkabaa, Ehsan Nazemi, Osman Taylan and El Mostafa Kalmoun

Mathematics 2021, 9(13), 1460; https://doi.org/10.3390/math9131460 - 22 Jun 2021

Cited by 6 | Viewed by 1826

Abstract

To the best knowledge of the authors, in former studies in the field of measuring volume fraction of gas, oil, and water components in a three-phase flow using gamma radiation technique, the existence of a scale layer has not been considered. The formed [...] Read more.

To the best knowledge of the authors, in former studies in the field of measuring volume fraction of gas, oil, and water components in a three-phase flow using gamma radiation technique, the existence of a scale layer has not been considered. The formed scale layer usually has a higher density in comparison to the fluid flow inside the oil pipeline, which can lead to high photon attenuation and, consequently, reduce the measuring precision of three-phase flow meter. The purpose of this study is to present an intelligent gamma radiation-based, nondestructive technique with the ability to measure volume fraction of gas, oil, and water components in the annular regime of a three-phase flow independent of the scale layer. Since, in this problem, there are several unknown parameters, such as gas, oil, and water components with different amounts and densities and scale layers with different thicknesses, it is not possible to measure the volume fraction using a conventional gamma radiation system. In this study, a system including a ²⁴¹Am-¹³³Ba dual energy source and two transmission detectors was used. The first detector was located diametrically in front of the source. For the second detector, at first, a sensitivity investigation was conducted in order to find the optimum position. The four extracted signals in both detectors (counts under photo peaks of both detectors) were used as inputs of neural network, and volume fractions of gas and oil components were utilized as the outputs. Using the proposed intelligent technique, volume fraction of each component was predicted independent of the barium sulfate scale layer, with a maximum MAE error of 3.66%. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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15 pages, 2145 KiB

Open AccessArticle

Acceleration of Boltzmann Collision Integral Calculation Using Machine Learning

by Ian Holloway, Aihua Wood and Alexander Alekseenko

Mathematics 2021, 9(12), 1384; https://doi.org/10.3390/math9121384 - 15 Jun 2021

Cited by 5 | Viewed by 2336

Abstract

The Boltzmann equation is essential to the accurate modeling of rarefied gases. Unfortunately, traditional numerical solvers for this equation are too computationally expensive for many practical applications. With modern interest in hypersonic flight and plasma flows, to which the Boltzmann equation is relevant, [...] Read more.

The Boltzmann equation is essential to the accurate modeling of rarefied gases. Unfortunately, traditional numerical solvers for this equation are too computationally expensive for many practical applications. With modern interest in hypersonic flight and plasma flows, to which the Boltzmann equation is relevant, there would be immediate value in an efficient simulation method. The collision integral component of the equation is the main contributor of the large complexity. A plethora of new mathematical and numerical approaches have been proposed in an effort to reduce the computational cost of solving the Boltzmann collision integral, yet it still remains prohibitively expensive for large problems. This paper aims to accelerate the computation of this integral via machine learning methods. In particular, we build a deep convolutional neural network to encode/decode the solution vector, and enforce conservation laws during post-processing of the collision integral before each time-step. Our preliminary results for the spatially homogeneous Boltzmann equation show a drastic reduction of computational cost. Specifically, our algorithm requires

O (n^{3})

operations, while asymptotically converging direct discretization algorithms require

O (n^{6})

, where n is the number of discrete velocity points in one velocity dimension. Our method demonstrated a speed up of 270 times compared to these methods while still maintaining reasonable accuracy. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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15 pages, 8138 KiB

Open AccessArticle

Application of Feature Extraction and Artificial Intelligence Techniques for Increasing the Accuracy of X-ray Radiation Based Two Phase Flow Meter

by Abdulrahman Basahel, Mohammad Amir Sattari, Osman Taylan and Ehsan Nazemi

Mathematics 2021, 9(11), 1227; https://doi.org/10.3390/math9111227 - 27 May 2021

Cited by 34 | Viewed by 2824

Abstract

The increasing consumption of fossil fuel resources in the world has placed emphasis on flow measurements in the oil industry. This has generated a growing niche in the flowmeter industry. In this regard, in this study, an artificial neural network (ANN) and various [...] Read more.

The increasing consumption of fossil fuel resources in the world has placed emphasis on flow measurements in the oil industry. This has generated a growing niche in the flowmeter industry. In this regard, in this study, an artificial neural network (ANN) and various feature extractions have been utilized to enhance the precision of X-ray radiation-based two-phase flowmeters. The detection system proposed in this article comprises an X-ray tube, a NaI detector to record the photons, and a Pyrex-glass pipe, which is placed between detector and source. To model the mentioned geometry, the Monte Carlo MCNP-X code was utilized. Five features in the time domain were derived from the collected data to be used as the neural network input. Multi-Layer Perceptron (MLP) was applied to approximate the function related to the input-output relationship. Finally, the introduced approach was able to correctly recognize the flow pattern and predict the volume fraction of two-phase flow’s components with root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) of less than 0.51, 0.4 and 1.16%, respectively. The obtained precision of the proposed system in this study is better than those reported in previous works. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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16 pages, 7599 KiB

Open AccessArticle

Defect Detection in Atomic Resolution Transmission Electron Microscopy Images Using Machine Learning

by Philip Cho, Aihua Wood, Krishnamurthy Mahalingam and Kurt Eyink

Mathematics 2021, 9(11), 1209; https://doi.org/10.3390/math9111209 - 27 May 2021

Cited by 7 | Viewed by 3660

Abstract

Point defects play a fundamental role in the discovery of new materials due to their strong influence on material properties and behavior. At present, imaging techniques based on transmission electron microscopy (TEM) are widely employed for characterizing point defects in materials. However, current [...] Read more.

Point defects play a fundamental role in the discovery of new materials due to their strong influence on material properties and behavior. At present, imaging techniques based on transmission electron microscopy (TEM) are widely employed for characterizing point defects in materials. However, current methods for defect detection predominantly involve visual inspection of TEM images, which is laborious and poses difficulties in materials where defect related contrast is weak or ambiguous. Recent efforts to develop machine learning methods for the detection of point defects in TEM images have focused on supervised methods that require labeled training data that is generated via simulation. Motivated by a desire for machine learning methods that can be trained on experimental data, we propose two self-supervised machine learning algorithms that are trained solely on images that are defect-free. Our proposed methods use principal components analysis (PCA) and convolutional neural networks (CNN) to analyze a TEM image and predict the location of a defect. Using simulated TEM images, we show that PCA can be used to accurately locate point defects in the case where there is no imaging noise. In the case where there is imaging noise, we show that incorporating a CNN dramatically improves model performance. Our models rely on a novel approach that uses the residual between a TEM image and its PCA reconstruction. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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15 pages, 1535 KiB

Open AccessArticle

A Conservative and Implicit Second-Order Nonlinear Numerical Scheme for the Rosenau-KdV Equation

by Cui Guo, Yinglin Wang and Yuesheng Luo

Mathematics 2021, 9(11), 1183; https://doi.org/10.3390/math9111183 - 24 May 2021

Cited by 4 | Viewed by 1490

Abstract

In this paper, for solving the nonlinear Rosenau-KdV equation, a conservative implicit two-level nonlinear scheme is proposed by a new numerical method named the multiple integral finite volume method. According to the order of the original differential equation’s highest derivative, we can confirm [...] Read more.

In this paper, for solving the nonlinear Rosenau-KdV equation, a conservative implicit two-level nonlinear scheme is proposed by a new numerical method named the multiple integral finite volume method. According to the order of the original differential equation’s highest derivative, we can confirm the number of integration steps, which is just called multiple integration. By multiple integration, a partial differential equation can be converted into a pure integral equation. This is very important because we can effectively avoid the large errors caused by directly approximating the derivative of the original differential equation using the finite difference method. We use the multiple integral finite volume method in the spatial direction and use finite difference in the time direction to construct the numerical scheme. The precision of this scheme is

O (τ^{2} + h^{3})

. In addition, we verify that the scheme possesses the conservative property on the original equation. The solvability, uniqueness, convergence, and unconditional stability of this scheme are also demonstrated. The numerical results show that this method can obtain highly accurate solutions. Further, the tendency of the numerical results is consistent with the tendency of the analytical results. This shows that the discrete scheme is effective. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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19 pages, 6093 KiB

Open AccessArticle

Numerical Investigation of MHD Pulsatile Flow of Micropolar Fluid in a Channel with Symmetrically Constricted Walls

by Amjad Ali, Muhammad Umar, Zaheer Abbas, Gullnaz Shahzadi, Zainab Bukhari and Arshad Saleem

Mathematics 2021, 9(9), 1000; https://doi.org/10.3390/math9091000 - 28 Apr 2021

Cited by 8 | Viewed by 1608

Abstract

This article presented an analysis of the pulsatile flow of non-Newtonian micropolar (MP) fluid under Lorentz force’s effect in a channel with symmetrical constrictions on the walls. The governing equations were first converted into the vorticity–stream function form, and a finite difference-based solver [...] Read more.

This article presented an analysis of the pulsatile flow of non-Newtonian micropolar (MP) fluid under Lorentz force’s effect in a channel with symmetrical constrictions on the walls. The governing equations were first converted into the vorticity–stream function form, and a finite difference-based solver was used to solve it numerically on a Cartesian grid. The impacts of different flow controlling parameters, including the Hartman number, Strouhal number, Reynolds number, and MP parameter on the flow profiles, were studied. The wall shear stress (WSS), axial, and micro-rotation velocity profiles were depicted visually. The streamlines and vorticity patterns of the flow were also sketched. It is evident from the numerical results that the flow separation region near constriction as well as flattening of the axial velocity component is effectively controlled by the Hartmann number. At the maximum flow rate, the WSS attained its peak. The WSS increased in both the Hartmann number and Reynolds number, whereas it declined with the higher values of the MP parameter. The micro-rotation velocity increased in the Reynolds number, and it declined with increment in the MP parameter. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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23 pages, 1150 KiB

Open AccessArticle

Meta-Heuristic Optimization Methods for Quaternion-Valued Neural Networks

by Jeremiah Bill, Lance Champagne, Bruce Cox and Trevor Bihl

Mathematics 2021, 9(9), 938; https://doi.org/10.3390/math9090938 - 23 Apr 2021

Cited by 5 | Viewed by 2179

Abstract

In recent years, real-valued neural networks have demonstrated promising, and often striking, results across a broad range of domains. This has driven a surge of applications utilizing high-dimensional datasets. While many techniques exist to alleviate issues of high-dimensionality, they all induce a cost [...] Read more.

In recent years, real-valued neural networks have demonstrated promising, and often striking, results across a broad range of domains. This has driven a surge of applications utilizing high-dimensional datasets. While many techniques exist to alleviate issues of high-dimensionality, they all induce a cost in terms of network size or computational runtime. This work examines the use of quaternions, a form of hypercomplex numbers, in neural networks. The constructed networks demonstrate the ability of quaternions to encode high-dimensional data in an efficient neural network structure, showing that hypercomplex neural networks reduce the number of total trainable parameters compared to their real-valued equivalents. Finally, this work introduces a novel training algorithm using a meta-heuristic approach that bypasses the need for analytic quaternion loss or activation functions. This algorithm allows for a broader range of activation functions over current quaternion networks and presents a proof-of-concept for future work. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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18 pages, 8567 KiB

Open AccessArticle

Hybrid Nanofluid Flow over a Permeable Non-Isothermal Shrinking Surface

by Iskandar Waini, Anuar Ishak and Ioan Pop

Mathematics 2021, 9(5), 538; https://doi.org/10.3390/math9050538 - 04 Mar 2021

Cited by 33 | Viewed by 1908

Abstract

In this paper, we examine the influence of hybrid nanoparticles on flow and heat transfer over a permeable non-isothermal shrinking surface and we also consider the radiation and the magnetohydrodynamic (MHD) effects. A hybrid nanofluid consists of copper (Cu) and alumina (Al₂ [...] Read more.

In this paper, we examine the influence of hybrid nanoparticles on flow and heat transfer over a permeable non-isothermal shrinking surface and we also consider the radiation and the magnetohydrodynamic (MHD) effects. A hybrid nanofluid consists of copper (Cu) and alumina (Al₂O₃) nanoparticles which are added into water to form Cu-Al₂O₃/water. The similarity equations are obtained using a similarity transformation and numerical results are obtained via bvp4c in MATLAB. The results show that dual solutions are dependent on the suction strength of the shrinking surface; in addition, the heat transfer rate is intensified with an increase in the magnetic parameter and the hybrid nanoparticles volume fractions for higher values of the radiation parameter. Furthermore, the heat transfer rate is higher for isothermal surfaces as compared with non-isothermal surfaces. Further analysis proves that the first solution is physically reliable and stable. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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29 pages, 1708 KiB

Open AccessArticle

Modeling and Simulation Techniques Used in High Strain Rate Projectile Impact

by Derek G. Spear, Anthony N. Palazotto and Ryan A. Kemnitz

Mathematics 2021, 9(3), 274; https://doi.org/10.3390/math9030274 - 30 Jan 2021

Cited by 4 | Viewed by 2830

Abstract

A series of computational models and simulations were conducted for determining the dynamic responses of a solid metal projectile impacting a target under a prescribed high strain rate loading scenario in three-dimensional space. The focus of this study was placed on two different [...] Read more.

A series of computational models and simulations were conducted for determining the dynamic responses of a solid metal projectile impacting a target under a prescribed high strain rate loading scenario in three-dimensional space. The focus of this study was placed on two different modeling techniques within finite element analysis available in the Abaqus software suite. The first analysis technique relied heavily on more traditional Lagrangian analysis methods utilizing a fixed mesh, while still taking advantage of the finite difference integration present under the explicit analysis approach. A symmetry reduced model using the Lagrangian coordinate system was also developed for comparison in physical and computational performance. The second analysis technique relied on a mixed model that still made use of some Lagrangian modeling, but included smoothed particle hydrodynamics techniques as well, which are mesh free. The inclusion of the smoothed particle hydrodynamics was intended to address some of the known issues in Lagrangian analysis under high displacement and deformation. A comparison of the models was first performed against experimental results as a validation of the models, then the models were compared against each other based on closeness to experimentation and computational performance. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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14 pages, 758 KiB

Open AccessArticle

Issues on Applying One- and Multi-Step Numerical Methods to Chaotic Oscillators for FPGA Implementation

by Omar Guillén-Fernández, María Fernanda Moreno-López and Esteban Tlelo-Cuautle

Mathematics 2021, 9(2), 151; https://doi.org/10.3390/math9020151 - 12 Jan 2021

Cited by 19 | Viewed by 2582

Abstract

Chaotic oscillators have been designed with embedded systems like field-programmable gate arrays (FPGAs), and applied in different engineering areas. However, the majority of works do not detail the issues when choosing a numerical method and the associated electronic implementation. In this manner, we [...] Read more.

Chaotic oscillators have been designed with embedded systems like field-programmable gate arrays (FPGAs), and applied in different engineering areas. However, the majority of works do not detail the issues when choosing a numerical method and the associated electronic implementation. In this manner, we show the FPGA implementation of chaotic and hyper-chaotic oscillators from the selection of a one-step or multi-step numerical method. We highlight that one challenge is the selection of the time-step h to increase the frequency of operation. The case studies include the application of three one-step and three multi-step numerical methods to simulate three chaotic and two hyper-chaotic oscillators. The numerical methods provide similar chaotic time-series, which are used within a time-series analyzer (TISEAN) to evaluate the Lyapunov exponents and Kaplan–Yorke dimension (

D_{K Y}

) of the (hyper-)chaotic oscillators. The oscillators providing higher exponents and

D_{K Y}

are chosen because higher values mean that the chaotic time series may be more random to find applications in chaotic secure communications. In addition, we choose representative numerical methods to perform their FPGA implementation, which hardware resources are described and counted. It is highlighted that the Forward Euler method requires the lowest hardware resources, but it has lower stability and exactness compared to other one-step and multi-step methods. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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12 pages, 837 KiB

Open AccessArticle

A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

by Benjamin Akers, Tony Liu and Jonah Reeger

Mathematics 2021, 9(1), 65; https://doi.org/10.3390/math9010065 - 30 Dec 2020

Cited by 3 | Viewed by 2039

Abstract

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on

R

, the former [...] Read more.

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on

R

, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed. Full article

(This article belongs to the Special Issue Applied Mathematics and Computational Physics)

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