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Volume 29
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Math. Comput. Appl., Volume 29, Issue 2 (April 2024) – 13 articles

Cover Story (view full-size image): This paper evaluates the local radial basis function collocation method using multiquadrics (MQs) and polyharmonic splines (PHSs) to solve steady and transient diffusion problems. Tests involve standard boundary and initial value cases with Dirichlet, Neumann, and Robin boundary conditions. A detailed analysis of node density, timestep, influence domain, node regularity, and shape parameters is performed. The advantages and drawbacks of using MQs and PHSs are outlined. View this paper
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24 pages, 8167 KiB  
Article
A Robust FOPD Controller That Allows Faster Detection of Defects for Touch Panels
by Yuan-Jay Wang
Math. Comput. Appl. 2024, 29(2), 29; https://doi.org/10.3390/mca29020029 - 16 Apr 2024
Viewed by 1031
Abstract
This study aims to synthesize and implement a robust fractional order PD (RFOPD) controller to increase the speed at which defects in automated touch panel inspection systems (ATPISs) are detected. A three-dimensional orthogonal stage (TDOS) driven by BLDC servo motors moves the inspection [...] Read more.
This study aims to synthesize and implement a robust fractional order PD (RFOPD) controller to increase the speed at which defects in automated touch panel inspection systems (ATPISs) are detected. A three-dimensional orthogonal stage (TDOS) driven by BLDC servo motors moves the inspection pen (IP) vertically and horizontally. The dynamic equation relating the BLDC servo motor input to the tip motion is established. A touch position identification (TPI) system is used to locate the touch point rapidly. An RFOPD controller is used to actuate the BLDC servo motors and move the TDOS rapidly and accurately in three dimensions. This method displaces the IP to any specified position and shows user-defined inspection trajectories on the touch screens. The gain-phase margin tester (GPMT) and stability equation methods are exploited to schedule the RFOPD controller gain settings and to maintain the specific safety margins for the controlled system. The simulation studies show that the proposed RFOPD controller exhibits better tracking and disturbance rejection responses than a conventional PID controller. The robustness of the RFOPD-controlled ATPIS, considering unmodeled uncertainties and friction-induced disturbances, is verified through simulation and experimental studies. Several user-defined inspection patterns are used to verify performance, and the experimental results show that the proposed RFOPD controller is effective. Full article
(This article belongs to the Topic Mathematical Modeling)
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<p>A photo of the homemade ATPIS.</p> Full article ">Figure 2
<p>The general structure of 5-wire resistive TPs.</p> Full article ">Figure 3
<p>The internal structure of general surface capacitive TPs.</p> Full article ">Figure 4
<p>The internal structure of general projected capacitive TPs.</p> Full article ">Figure 5
<p>Three-dimensional inspection pen control system.</p> Full article ">Figure 6
<p>The vertical translation Z-stage with a compact linear actuator.</p> Full article ">Figure 7
<p>The GUICS for the ATPIS.</p> Full article ">Figure 8
<p>Electro-mechanical photograph of the SABSD stage.</p> Full article ">Figure 9
<p>Schematic diagram of the SABSD stage.</p> Full article ">Figure 10
<p>Schematic diagram of an SABSD system driven by a BLDC servo motor.</p> Full article ">Figure 11
<p>A simplified block diagram of the FOPD-controlled SABSD system with a GMPT.</p> Full article ">Figure 12
<p>The 5 dB and 30 Deg boundaries, the stability boundary, and the FSOR(<span class="html-italic">K<sub>P</sub></span>, <span class="html-italic">K<sub>D</sub></span>) region.</p> Full article ">Figure 13
<p>The FSOR(<span class="html-italic">K<sub>P</sub></span>, <span class="html-italic">K<sub>D</sub></span>) region for −10%, +0%, and +10% LG variations, the RFSOR(<span class="html-italic">K<sub>P</sub></span>, <span class="html-italic">K<sub>D</sub></span>) region, the CIU region, and the optimal RFOPD controller, P1(<span class="html-italic">K<sub>P</sub></span><sub>1</sub> = 37,000, <span class="html-italic">K<sub>D</sub></span><sub>1</sub> = 4300).</p> Full article ">Figure 14
<p>The unit step and disturbance responses subject to −10%, +0%, and +10% LG variations: (<b>a</b>) Case C; (<b>b</b>) Case B; (<b>c</b>) Case A.</p> Full article ">Figure 15
<p>Simulation studies for a round-type inspection: (<b>a</b>) Case A: RFOPD controller; (<b>b</b>) Case B: PD controller.</p> Full article ">Figure 16
<p>Diagonal-line inspection: (<b>a</b>) passed and (<b>b</b>) failed.</p> Full article ">Figure 17
<p>Rectangular-type inspection: (<b>a</b>) passed and (<b>b</b>) failed.</p> Full article ">Figure 18
<p>Circular-type inspection: (<b>a</b>) passed and (<b>b</b>) failed.</p> Full article ">Figure 19
<p>Convergent-type inspection: (<b>a</b>) passed and (<b>b</b>) failed.</p> Full article ">Figure 20
<p>Round-type inspection: (<b>a</b>) passed and (<b>b</b>) failed.</p> Full article ">Figure 21
<p>Rhombus-type inspection: (<b>a</b>) passed and (<b>b</b>) failed.</p> Full article ">
16 pages, 293 KiB  
Article
Global Behavior of Solutions to a Higher-Dimensional System of Difference Equations with Lucas Numbers Coefficients
by Messaoud Berkal, Juan Francisco Navarro and Raafat Abo-Zeid
Math. Comput. Appl. 2024, 29(2), 28; https://doi.org/10.3390/mca29020028 - 31 Mar 2024
Cited by 1 | Viewed by 1417
Abstract
In this paper, we derive the well-defined solutions to a θ-dimensional system of difference equations. We show that, the well-defined solutions to that system are represented in terms of Fibonacci and Lucas sequences. Moreover, we study the global stability of the solutions [...] Read more.
In this paper, we derive the well-defined solutions to a θ-dimensional system of difference equations. We show that, the well-defined solutions to that system are represented in terms of Fibonacci and Lucas sequences. Moreover, we study the global stability of the solutions to that system. Finally, we give some numerical examples which confirm our theoretical results. Full article
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<p>A graph representing the global stability of system (<a href="#FD15-mca-29-00028" class="html-disp-formula">15</a>) (<b>left</b>) and system (<a href="#FD16-mca-29-00028" class="html-disp-formula">16</a>) (<b>right</b>).</p> Full article ">
14 pages, 1926 KiB  
Article
Modeling of Chemical Vapor Infiltration for Fiber-Reinforced Silicon Carbide Composites Using Meshless Method of Fundamental Solutions
by Patrick Mahoney and Alex Povitsky
Math. Comput. Appl. 2024, 29(2), 27; https://doi.org/10.3390/mca29020027 - 22 Mar 2024
Cited by 1 | Viewed by 1553
Abstract
In this study, the Method of Fundamental Solutions (MFSs) is adopted to model Chemical Vapor Infiltration (CVI) in a fibrous preform. The preparation of dense fiber-reinforced silicon carbide composites is considered. The reaction flux at the solid surface is equal to the diffusion [...] Read more.
In this study, the Method of Fundamental Solutions (MFSs) is adopted to model Chemical Vapor Infiltration (CVI) in a fibrous preform. The preparation of dense fiber-reinforced silicon carbide composites is considered. The reaction flux at the solid surface is equal to the diffusion flux towards the surface. The Robin or third-type boundary condition is implemented into the MFS. From the fibers’ surface concentrations obtained by MFS, deposition rates are calculated, and the geometry is updated at each time step, modeling the pore filling over time. The MFS solution is verified by comparing the results to a known analytical solution for a simplified geometry of concentric cylinders with a concentration set at the outer cylinder and a reaction at the inner cylinder. MFS solutions are compared to published experimental data. Porosity transients are obtained by a combination of MFSs with surface deposition to show the relation between the initial and final porosities. Full article
(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)
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<p>Infiltration for randomly placed fibers with aligned axis: (<b>a</b>) initial geometry of fibers and (<b>b</b>) final geometry after the deposition process has concluded. The deposited layer (grey) and fibers (black) are shown.</p> Full article ">Figure 2
<p>The set up of diffusion between two cylinders with surface reaction at the inner cylinder. Concentration <span class="html-italic">C</span> at the outer cylinder surface is given. The inner cylinder surface reaction has a reaction boundary at equilibrium.</p> Full article ">Figure 3
<p>The MFS set up for diffusion between two cylinders is as follows: red markers are submerged singularities, while blue markers are observation points.</p> Full article ">Figure 4
<p>Comparison of the concentration <span class="html-italic">c</span>(<span class="html-italic">r</span>) obtained by MFS with the analytical solution (14).</p> Full article ">Figure 5
<p>Example of void (red) in preform of fibers with deposited material (blue).</p> Full article ">Figure 6
<p>Comparison of infiltrated geometry (<b>a</b>) computed by MFS and (<b>b</b>) obtained experimentally, see [<a href="#B35-mca-29-00027" class="html-bibr">35</a>], <a href="#mca-29-00027-f004" class="html-fig">Figure 4</a>, reprinted with Elsevier’s permission.</p> Full article ">
18 pages, 1253 KiB  
Article
Testing Homogeneity of Proportion Ratios for Stratified Bilateral Correlated Data
by Wanqing Tian and Changxing Ma
Math. Comput. Appl. 2024, 29(2), 26; https://doi.org/10.3390/mca29020026 - 22 Mar 2024
Viewed by 1363
Abstract
Intraclass correlation in bilateral data has been investigated in recent decades with various statistical methods. In practice, stratifying bilateral data by some control variables will provide more sophisticated statistical results to satisfy different research proposed in randomized clinical trials. In this article, we [...] Read more.
Intraclass correlation in bilateral data has been investigated in recent decades with various statistical methods. In practice, stratifying bilateral data by some control variables will provide more sophisticated statistical results to satisfy different research proposed in randomized clinical trials. In this article, we propose three test statistics (the likelihood ratio test, score test, and Wald-type test statistics) to evaluate the homogeneity of proportion ratios for stratified bilateral correlated data under an equal correlation assumption. Monte Carlo simulations of Type I error and power are performed, and the score test yields a robust outcome based on empirical Type I error and power. Lastly, two real data examples are conducted to illustrate the proposed three tests. Full article
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<p>Violin plots and box plots of empirical sizes (<math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>).</p> Full article ">Figure 2
<p>Violin plots and box plots of empirical sizes (<math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>).</p> Full article ">Figure 3
<p>Violin plots and box plots of empirical sizes (<math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>).</p> Full article ">Figure 4
<p>Violin plots and box plots of empirical sizes (<math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>).</p> Full article ">Figure 5
<p>Line plots of empirical power (<math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>).</p> Full article ">Figure 6
<p>Line plots of empirical power (<math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>).</p> Full article ">Figure 7
<p>Line plots of empirical power (<math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>).</p> Full article ">Figure 8
<p>Line plots of empirical power (<math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>).</p> Full article ">
23 pages, 6500 KiB  
Article
M5GP: Parallel Multidimensional Genetic Programming with Multidimensional Populations for Symbolic Regression
by Luis Cárdenas Florido, Leonardo Trujillo, Daniel E. Hernandez and Jose Manuel Muñoz Contreras
Math. Comput. Appl. 2024, 29(2), 25; https://doi.org/10.3390/mca29020025 - 18 Mar 2024
Cited by 2 | Viewed by 2696
Abstract
Machine learning and artificial intelligence are growing in popularity thanks to their ability to produce models that exhibit unprecedented performance in domains that include computer vision, natural language processing and code generation. However, such models tend to be very large and complex and [...] Read more.
Machine learning and artificial intelligence are growing in popularity thanks to their ability to produce models that exhibit unprecedented performance in domains that include computer vision, natural language processing and code generation. However, such models tend to be very large and complex and impossible to understand using traditional analysis or human scrutiny. Conversely, Symbolic Regression methods attempt to produce models that are relatively small and (potentially) human-readable. In this domain, Genetic Programming (GP) has proven to be a powerful search strategy that achieves state-of-the-art performance. This paper presents a new GP-based feature transformation method called M5GP, which is hybridized with multiple linear regression to produce linear models, implemented to exploit parallel processing on graphical processing units for efficient computation. M5GP is the most recent variant from a family of feature transformation methods (M2GP, M3GP and M4GP) that have proven to be powerful tools for both classification and regression tasks applied to tabular data. The proposed method was evaluated on SRBench v2.0, the current standard benchmarking suite for Symbolic Regression. Results show that M5GP achieves performance that is competitive with the state-of-the-art, achieving a top-three rank on the most difficult subset of black-box problems. Moreover, it achieves the lowest computation time when compared to other GP-based methods that have similar accuracy scores. Full article
(This article belongs to the Special Issue New Trends in Computational Intelligence and Applications 2023)
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<p>Wrapper-based processing for feature transformation with GP.</p> Full article ">Figure 2
<p>A set of individuals generated randomly in M5GP, each one on a single GPU thread using a Numba kernel.</p> Full article ">Figure 3
<p>Interpreter and stack-based processing in M5GP based on [<a href="#B23-mca-29-00025" class="html-bibr">23</a>] of the first individual in <a href="#mca-29-00025-f002" class="html-fig">Figure 2</a>. The variable values used in the example are: <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold">X</mi> <mn mathvariant="bold">1</mn> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold">X</mi> <mn mathvariant="bold">2</mn> </msub> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <msub> <mi mathvariant="bold">X</mi> <mn mathvariant="bold">3</mn> </msub> </semantics></math> = 1, <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold">X</mi> <mn mathvariant="bold">4</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold">X</mi> <mn mathvariant="bold">5</mn> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Parallel processing of the training data to generate the semantic matrix of each individual transformation model.</p> Full article ">Figure 5
<p>SRBench performance on the black-box regression problems for all M5GP variants in <a href="#mca-29-00025-t001" class="html-table">Table 1</a>: (<b>a</b>) <math display="inline"><semantics> <msup> <mi>R</mi> <mn>2</mn> </msup> </semantics></math> test set performance, (<b>b</b>) model size (functions and terminals in the model), and (<b>c</b>) total runtime during training. (*) Represents the method returns a symbolic model.</p> Full article ">Figure 6
<p>SRBench performance of M5GP compared to other methods, considering configuration M5GP-1, M5GP-2, M5GP-3 and M5GP-4: (<b>a</b>) <math display="inline"><semantics> <msup> <mi>R</mi> <mn>2</mn> </msup> </semantics></math> test set performance, (<b>b</b>) model size (terms and operators in the model), and (<b>c</b>) total runtime during training. (*) Represents the method returns a symbolic model.</p> Full article ">Figure 7
<p>Wilcoxon signed-rank test comparisons from SRBench considering M5GP-1. (<b>a</b>) <math display="inline"><semantics> <msup> <mi>R</mi> <mn>2</mn> </msup> </semantics></math>. (<b>b</b>) Model size.</p> Full article ">Figure 8
<p>Performance of M5GP on the black-box problems, showing improved performance on the Friedman subset of problems.</p> Full article ">Figure 9
<p>SRBench performance of M5GP on the ground-truth problems: (<b>a</b>) accuracy based on <math display="inline"><semantics> <msup> <mi>R</mi> <mn>2</mn> </msup> </semantics></math> and (<b>b</b>) solution rate given in percentage. Results are shown for both datasets used, Feynman and Strogatz.</p> Full article ">Figure 10
<p>Training time relative to: (<b>a</b>) the number of samples (<math display="inline"><semantics> <mrow> <mi>N</mi> <mi>s</mi> <mi>a</mi> <mi>m</mi> <mi>p</mi> <mi>l</mi> <mi>e</mi> <mi>s</mi> </mrow> </semantics></math>) used during training (<b>b</b>), the number of features (<math display="inline"><semantics> <mrow> <mi>N</mi> <mi>f</mi> <mi>e</mi> <mi>a</mi> <mi>t</mi> <mi>u</mi> <mi>r</mi> <mi>e</mi> <mi>s</mi> </mrow> </semantics></math>); and (<b>c</b>) the total size of the problem (<math display="inline"><semantics> <mrow> <mi>N</mi> <mi>p</mi> <mi>o</mi> <mi>i</mi> <mi>n</mi> <mi>t</mi> <mi>s</mi> <mo>=</mo> <mi>N</mi> <mi>s</mi> <mi>a</mi> <mi>m</mi> <mi>p</mi> <mi>l</mi> <mi>e</mi> <mi>s</mi> <mo>×</mo> <mi>N</mi> <mi>f</mi> <mi>e</mi> <mi>a</mi> <mi>t</mi> <mi>u</mi> <mi>r</mi> <mi>e</mi> <mi>s</mi> </mrow> </semantics></math>).</p> Full article ">
24 pages, 6398 KiB  
Article
A Coupled Finite-Boundary Element Method for Efficient Dynamic Structure-Soil-Structure Interaction Modeling
by Parham Azhir, Jafar Asgari Marnani, Mehdi Panji and Mohammad Sadegh Rohanimanesh
Math. Comput. Appl. 2024, 29(2), 24; https://doi.org/10.3390/mca29020024 - 18 Mar 2024
Cited by 3 | Viewed by 1959
Abstract
This paper introduces an innovative approach to numerically model Structure–Soil-Structure Interaction (SSSI) by integrating the Boundary Element Method (BEM) and the Finite Element Method (FEM) in a coupled manner. To assess the accuracy of the proposed method, a comparative study is undertaken, comparing [...] Read more.
This paper introduces an innovative approach to numerically model Structure–Soil-Structure Interaction (SSSI) by integrating the Boundary Element Method (BEM) and the Finite Element Method (FEM) in a coupled manner. To assess the accuracy of the proposed method, a comparative study is undertaken, comparing its outcomes with those generated by the conventional FEM technique. Alongside accuracy, the computational efficiency aspect is crucial for the analysis of large-scale SSSI problems. Hence, the computational performance of the coupled BEM–FEM method undergoes a thorough examination and is compared with that of the standalone FEM method. The results from these comparisons illustrate the superior capabilities of the proposed method in comparison to the FEM method. The novel approach provides more reliable results compared to traditional FEM methods, serving as a valuable tool for engineers and researchers involved in structural analysis and design. Full article
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<p>Displacement versus time in an arbitrary transient displacement.</p> Full article ">Figure 2
<p>Schematic expression of FEM-hosted BEM–FEM coupling approach.</p> Full article ">Figure 3
<p>Schematic representation of the first SSSI problem model.</p> Full article ">Figure 4
<p>The initial or primary applied load for the evaluation problem.</p> Full article ">Figure 5
<p>The subsequent or secondary applied load for the evaluation problem.</p> Full article ">Figure 6
<p>Elements utilized in the fem model for the initial problem.</p> Full article ">Figure 7
<p>A schematic illustration of the constituent elements encompassed in the model developed through the proposed method for the evaluation problem.</p> Full article ">Figure 8
<p>Displacement comparison of the first node in the first structure during initial loading.</p> Full article ">Figure 9
<p>Displacement comparison of the first node in the first structure during second loading.</p> Full article ">Figure 10
<p>Displacement comparison of the first node in the first structure during initial loading in unitless domain.</p> Full article ">Figure 11
<p>Displacement comparison of the first node in the first structure during second loading in unitless domain.</p> Full article ">Figure 12
<p>Deformation results of the soil at various distances from the center of the structure during the first loading phase of the problem in 1 hertz frequency.</p> Full article ">Figure 13
<p>Models of the structures with varying inter-structure distances.</p> Full article ">Figure 14
<p>Loading configuration throughout the analysis duration.</p> Full article ">Figure 15
<p>Deformation patterns over time with varied inter-structural spacings in the first structure.</p> Full article ">Figure 16
<p>Influence of the second structure on deformation characteristics of the first structure in the dimensionless domain (<math display="inline"><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math>).</p> Full article ">Figure 17
<p>Comparative analysis of deformation patterns over time with varied inter-structural spacings for the second structure.</p> Full article ">Figure 18
<p>Comparison of deformation patterns within the <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> domain across varied inter-structural spacings for the second structure.</p> Full article ">Figure 19
<p>Detailed comparative analysis of roof deformation patterns in the first structure under medium soil conditions.</p> Full article ">Figure 20
<p>Impact of the second structure on deformation characteristics of the first structure in the dimensionless domain (<math display="inline"><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math>) under medium soil conditions.</p> Full article ">Figure 21
<p>Comparative assessment of deformation patterns over time with varied inter-structural spacings in the second structure under medium soil conditions.</p> Full article ">Figure 22
<p>Comparative assessment of deformation patterns in the <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> domain with varied inter-structural spacings in the second structure in a medium soil environment.</p> Full article ">Figure 23
<p>Comparative assessment of deformation patterns over time with varied inter-structural spacings in the first structure under soft soil conditions.</p> Full article ">Figure 24
<p>Influence of the second structure on deformation characteristics of the first structure in the dimensionless domain (<math display="inline"><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math>) under soft soil conditions.</p> Full article ">Figure 25
<p>Comparative assessment of deformation patterns in the temporal domain with varied inter-structural spacings in the second structure, emphasizing the influence of soft soil conditions.</p> Full article ">Figure 26
<p>Comparative assessment of deformation patterns in the <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> domain for the second structure, considering diverse inter-structural spacings in a soft soil setting.</p> Full article ">
30 pages, 10410 KiB  
Article
Assessment of Local Radial Basis Function Collocation Method for Diffusion Problems Structured with Multiquadrics and Polyharmonic Splines
by Izaz Ali, Umut Hanoglu, Robert Vertnik and Božidar Šarler
Math. Comput. Appl. 2024, 29(2), 23; https://doi.org/10.3390/mca29020023 - 17 Mar 2024
Cited by 4 | Viewed by 1676
Abstract
This paper aims to systematically assess the local radial basis function collocation method, structured with multiquadrics (MQs) and polyharmonic splines (PHSs), for solving steady and transient diffusion problems. The boundary value test involves a rectangle with Dirichlet, Neuman, and Robin boundary conditions, and [...] Read more.
This paper aims to systematically assess the local radial basis function collocation method, structured with multiquadrics (MQs) and polyharmonic splines (PHSs), for solving steady and transient diffusion problems. The boundary value test involves a rectangle with Dirichlet, Neuman, and Robin boundary conditions, and the initial value test is associated with the Dirichlet jump problem on a square. The spectra of the free parameters of the method, i.e., node density, timestep, shape parameter, etc., are analyzed in terms of the average error. It is found that the use of MQs is less stable compared to PHSs for irregular node arrangements. For MQs, the most suitable shape parameter is determined for multiple cases. The relationship of the shape parameter with the total number of nodes, average error, node scattering factor, and the number of nodes in the local subdomain is also provided. For regular node arrangements, MQs produce slightly more accurate results, while for irregular node arrangements, PHSs provide higher accuracy than MQs. PHSs are recommended for use in diffusion problems that require irregular node spacing. Full article
(This article belongs to the Special Issue Radial Basis Functions)
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Figure 1
<p>Scheme of the domain Ω with boundary conditions weighted at Γ<sup>D</sup>, Γ<sup>R</sup>, and Γ<sup>N</sup>. The solid and empty circles show the interior and boundary nodes, respectively. The solid circular line shows the limits of the local sub-domain <sub>l</sub>Ω containing nine interior nodes. In contrast, a dashed circular line represents another local sub-domain, containing a boundary node and eight interior nodes, whereas the solid triangle shows the central node. <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>r</mi> <mi>max</mi> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>r</mi> <mi>min</mi> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> </mrow> </semantics></math> are the maximum and minimum distance between any node in the subdomain <math display="inline"><semantics> <mi>l</mi> </semantics></math>, respectively.</p> Full article ">Figure 2
<p>Local sub-domain scheme for RND with <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mo> </mo> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>9</mn> <mo>,</mo> <mo> </mo> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>13</mn> <mo>,</mo> <mo> </mo> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>25</mn> </mrow> </semantics></math>. Solid and empty circles denote the inner and boundary nodes, respectively.</p> Full article ">Figure 3
<p>Scheme of Case 1 with geometry (<math display="inline"><semantics> <mrow> <msubsup> <mi>p</mi> <mi>x</mi> <mo>−</mo> </msubsup> <mo>=</mo> <msubsup> <mi>p</mi> <mi>y</mi> <mo>−</mo> </msubsup> <mo>=</mo> <mn>0</mn> <mrow> <mo> </mo> <mi mathvariant="normal">m</mi> </mrow> <mo>,</mo> <mo> </mo> <msubsup> <mi>p</mi> <mi>x</mi> <mo>+</mo> </msubsup> <mo>=</mo> <mn>0.6</mn> <mo> </mo> <mi>m</mi> <mo>,</mo> <mo> </mo> <msubsup> <mi>p</mi> <mi>y</mi> <mo>+</mo> </msubsup> <mo>=</mo> <mn>1</mn> <mo> </mo> <mi>m</mi> </mrow> </semantics></math>) and boundary conditions.</p> Full article ">Figure 4
<p>Case 1, RND and QUND with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>13</mn> <mo>×</mo> <mn>21</mn> </mrow> </semantics></math> nodes. In regular node distribution, the <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>r</mi> <mi>min</mi> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> </mrow> </semantics></math> is 0.05 m and <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>r</mi> <mi>max</mi> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> </mrow> </semantics></math> is 0.1802 m, while for QUND, <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>r</mi> <mi>min</mi> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> </mrow> </semantics></math> is 0.0994 m, and the <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>r</mi> <mi>max</mi> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> </mrow> </semantics></math> is 0.1839 m.</p> Full article ">Figure 5
<p>Case 1, convergence analysis of analytical solution as a function of the terms <math display="inline"><semantics> <mi>i</mi> </semantics></math> used in the evaluation of Equation (33) for four different nodes, i.e., (0.6 m, 0.1 m), (0.5 m, 1 m), (0.3 m, 0.5 m), and (0 m, 0.9 m) of the rectangular geometry.</p> Full article ">Figure 6
<p>Scheme of Case 2 with geometry (<math display="inline"><semantics> <mrow> <msubsup> <mi>p</mi> <mi>x</mi> <mo>−</mo> </msubsup> <mo>=</mo> <msubsup> <mi>p</mi> <mi>y</mi> <mo>−</mo> </msubsup> <mo>=</mo> <mn>0</mn> <mo> </mo> <mi mathvariant="normal">m</mi> <mo>,</mo> <mo> </mo> <msubsup> <mi>p</mi> <mi>x</mi> <mo>+</mo> </msubsup> <mo>=</mo> <msubsup> <mi>p</mi> <mi>y</mi> <mo>+</mo> </msubsup> <mo>=</mo> <mn>1</mn> <mo> </mo> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>) and boundary conditions.</p> Full article ">Figure 7
<p>Case 2, regular nodes distribution is shown for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>21</mn> <mo>×</mo> <mn>21</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>13</mn> </mrow> </semantics></math> node arrangement with <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>r</mi> <mi>min</mi> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> </mrow> </semantics></math> is 0.05 m and <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>r</mi> <mi>max</mi> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> </mrow> </semantics></math> is 0.18027 m, solid and empty circles representing the inner and boundary nodes respectively.</p> Full article ">Figure 8
<p>Case 2, convergence analysis of analytical solution as a function of the terms <math display="inline"><semantics> <mi>i</mi> </semantics></math> used in evaluating Equation (39) for two nodes, i.e., at (0.5 m, 0.5 m) and (0.6 m, 0.9 m), of the square geometry for time <span class="html-italic">t</span> = 0.001 s, 0.1 s, and 1 s.</p> Full article ">Figure 9
<p>Case 1, the difference in % of the <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of the node distance for MQ with and without augmentation (RND, <span class="html-italic">c</span> = 64, <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>).</p> Full article ">Figure 10
<p>Case 1, improvement in % of the <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of the node distance for MQ with and without augmentation (RND, <span class="html-italic">c</span> = 64, <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>).</p> Full article ">Figure 11
<p>Case 1, improvement in % of the <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of the node distance for MQ with and without augmentation (RND, <span class="html-italic">c</span> = 32, <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>13</mn> </mrow> </semantics></math>).</p> Full article ">Figure 12
<p>Case 1, MQ, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of node distance calculated for five different shape pa-rameters (RND, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>).</p> Full article ">Figure 13
<p>Case 1, MQ, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of shape parameter (RND, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>).</p> Full article ">Figure 14
<p>Case 1, MQ, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of the node distance for different shape parameters (RND, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>13</mn> </mrow> </semantics></math>).</p> Full article ">Figure 15
<p>Case 1, MQ, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of the shape parameter (RND, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>13</mn> </mrow> </semantics></math>).</p> Full article ">Figure 16
<p>Case 1, PHS, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of node distance for a different number of nodes in the local subdomain (RND, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>).</p> Full article ">Figure 17
<p>Case 1, MQ, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function node distance with a different number of nodes in the local subdomain and optimum shape parameter (RND, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p> Full article ">Figure 18
<p>Case 1, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of node distance (RND, MQ with <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and PHS with <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>).</p> Full article ">Figure 19
<p>Case 1, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of the scattering factor (<math display="inline"><semantics> <mi>δ</mi> </semantics></math>) (QUND, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>241</mn> <mo>×</mo> <mn>401</mn> </mrow> </semantics></math>).</p> Full article ">Figure 20
<p>Case 1, PHS, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> error as a function of the node distance for different <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> </mrow> </semantics></math> (QUND, <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>0.10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>).</p> Full article ">Figure 21
<p>Case 1, MQ, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of the node distance for different <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> </mrow> </semantics></math> (QUND, <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>0.10</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p> Full article ">Figure 22
<p>Case 1, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of node distance (QUND, <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>0.10</mn> </mrow> </semantics></math>, MQ with <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, PHS with <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>).</p> Full article ">Figure 23
<p>Case 1, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of the shape parameter (c) for scaled and unscaled MQs (RND, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>241</mn> <mo>×</mo> <mn>401</mn> </mrow> </semantics></math>).</p> Full article ">Figure 24
<p>Case 1, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of the scattering factor (<math display="inline"><semantics> <mi>δ</mi> </semantics></math>) for scaled and unscaled MQs (QUND, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>241</mn> <mo>×</mo> <mn>401</mn> </mrow> </semantics></math>).</p> Full article ">Figure 25
<p>Case 2, MQ, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of node distance for four different times (RND, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>13</mn> </mrow> </semantics></math>, <span class="html-italic">c</span> = 32).</p> Full article ">Figure 26
<p>Case 2, MQ, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of node distance for three different <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> </mrow> </semantics></math> (RND, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <span class="html-italic">c</span> = 32, <span class="html-italic">t</span> = 1 s).</p> Full article ">Figure 27
<p>Case 2, MQ, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of node distance for four different <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> </mrow> </semantics></math> with optimal shape parameters (RND, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <span class="html-italic">t</span> = 1 s).</p> Full article ">Figure 28
<p>Case 2, PHS, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of node distance for four different times (RND, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>).</p> Full article ">Figure 29
<p>Case 2, PHS, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of node distance for four different times (RND, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>13</mn> </mrow> </semantics></math>).</p> Full article ">Figure 30
<p>Case 2, PHS, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of node distance for four different times and three node arrangements (RND, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>25</mn> </mrow> </semantics></math>).</p> Full article ">Figure 31
<p>Case 2, PHS, <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of node distance for three different <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> </mrow> </semantics></math> (RND, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <span class="html-italic">t</span> = 1 s,).</p> Full article ">Figure 32
<p>Case 2, comparison of MQ (<math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>) and PHS (<math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>a</mi> <mi>u</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>) in terms of <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mi>v</mi> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math> as a function of node distance (RND, <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>N</mi> <none/> <none/> <mprescripts/> <mi>l</mi> <none/> </mmultiscripts> <mo>=</mo> <mn>13</mn> </mrow> </semantics></math>, <span class="html-italic">t</span> = 1 s).</p> Full article ">
19 pages, 2491 KiB  
Article
Variability on Functionally Graded Plates’ Deflection Due to Uncertainty on Carbon Nanotubes’ Properties
by Alda Carvalho, Ana Martins, Ana F. Mota and Maria A. R. Loja
Math. Comput. Appl. 2024, 29(2), 22; https://doi.org/10.3390/mca29020022 - 16 Mar 2024
Cited by 2 | Viewed by 1300
Abstract
Carbon nanotubes are widely used as material reinforcement in diverse fields of engineering. Being that their contribution is significant to improving the mean properties of the resulting materials, it is important to assess the influence of the variability on carbon nanotubes’ material and [...] Read more.
Carbon nanotubes are widely used as material reinforcement in diverse fields of engineering. Being that their contribution is significant to improving the mean properties of the resulting materials, it is important to assess the influence of the variability on carbon nanotubes’ material and geometrical properties to structures’ responses. This work considers functionally graded plates constituted by an aluminum continuous phase reinforced with single-walled or multi-walled carbon. The nanotubes' weight fraction evolution through the thickness is responsible for the plates’ functional gradient. The plates’ samples are simulated considering that only the nanotubes’ material and geometrical characteristics are affected by uncertainty. The results obtained from the multiple regression models developed allow us to conclude that the length of the nanotubes has no impact on the maximum transverse displacement of the plates in opposition to the carbon nanotubes’ weight fraction evolution, their internal and external diameters, and the Young’s modulus. The multiple regression models developed can be used as alternative prediction tools within the domain of the study. Full article
(This article belongs to the Special Issue Mathematical and Computational Approaches in Applied Mechanics: A Themed Issue Dedicated to Professor J.N. Reddy)
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<p>Schematic representation of the graded mixture of carbon nanotubes and aluminum through plates’ thickness.</p> Full article ">Figure 2
<p>Illustrative flowchart of the simulation process.</p> Full article ">Figure 3
<p>Matrix plot for the maximum transverse displacement, (<math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>), and all the input variables.</p> Full article ">Figure 4
<p>Plates’ maximum transverse displacement [m] for the different values of the power law’ exponent and CNT type.</p> Full article ">Figure 5
<p>Matrix plot for the plates’ maximum transverse displacement and all the input parameters: Plates with SWCNT.</p> Full article ">Figure 6
<p>Matrix plot for the plates’ maximum transverse displacement and all the input parameters: Plates with MWCNT.</p> Full article ">
22 pages, 1823 KiB  
Article
Semi-Supervised Machine Learning Method for Predicting Observed Individual Risk Preference Using Gallup Data
by Faroque Ahmed, Mrittika Shamsuddin, Tanzila Sultana and Rittika Shamsuddin
Math. Comput. Appl. 2024, 29(2), 21; https://doi.org/10.3390/mca29020021 - 15 Mar 2024
Cited by 1 | Viewed by 2141
Abstract
Risk and uncertainty play a vital role in almost every significant economic decision, and an individual’s propensity to make riskier decisions also depends on various circumstances. This article aims to investigate the effects of social and economic covariates on an individual’s willingness to [...] Read more.
Risk and uncertainty play a vital role in almost every significant economic decision, and an individual’s propensity to make riskier decisions also depends on various circumstances. This article aims to investigate the effects of social and economic covariates on an individual’s willingness to take general risks and extends the scope of existing works by using quantitative measures of risk-taking from the GPS and Gallup datasets (in addition to the qualitative measures used in the literature). Based on the available observed risk-taking data for one year, this article proposes a semi-supervised machine learning-based approach that can efficiently predict the observed risk index for those countries/individuals for years when the observed risk-taking index was not collected. We find that linear models are insufficient to capture certain patterns among risk-taking factors, and non-linear models, such as random forest regression, can obtain better root mean squared values than those reported in past literature. In addition to finding factors that agree with past studies, we also find that subjective well-being influences risk-taking behavior. Full article
(This article belongs to the Section Social Sciences)
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<p>An overview of the research methodology used to predict ORP values for the Gallup data.</p> Full article ">Figure 2
<p>An overview of the data usage and the contributions of this research work.</p> Full article ">Figure 3
<p>Overview of the evaluation schemes used to evaluate the benchmark and semi-supervised models.</p> Full article ">Figure 4
<p>A schematic overview of the self-training semi-supervised learning mechanism using base learners. These base learners can be linear (e.g., find linear patterns only) or non-linear models (e.g., finds linear and non-linear patterns).</p> Full article ">Figure 5
<p>Feature sets for <math display="inline"><semantics> <msub> <mi>D</mi> <mrow> <mi>E</mi> <mi>x</mi> <mi>p</mi> <mi>e</mi> <mi>r</mi> <mi>t</mi> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>D</mi> <mrow> <mi>P</mi> <mi>o</mi> <mi>s</mi> <mi>s</mi> <mi>i</mi> <mi>b</mi> <mi>l</mi> <mi>e</mi> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>D</mi> <mrow> <mi>C</mi> <mi>o</mi> <mi>m</mi> <mi>p</mi> <mi>E</mi> <mi>x</mi> </mrow> </msub> </semantics></math>.</p> Full article ">Figure 6
<p>Benchmark LR Model Evaluation. Independent variables are chosen by the human expert.</p> Full article ">Figure 7
<p>Scatter plot comparison for LR Self-training, first iteration, Step 1 (<b>left</b>) and Step 2 (<b>right</b>). This particular figure uses the <math display="inline"><semantics> <msub> <mi>D</mi> <mrow> <mi>E</mi> <mi>x</mi> <mi>p</mi> <mi>e</mi> <mi>r</mi> <mi>t</mi> </mrow> </msub> </semantics></math> dataset.</p> Full article ">Figure 8
<p>The RMSE values of the four ML regression models over the ten iterations of the self-training stage plus the initially supervised base learner.</p> Full article ">Figure 9
<p>The performance of the LR and RFR model on the three datasets, <math display="inline"><semantics> <msub> <mi>D</mi> <mrow> <mi>E</mi> <mi>x</mi> <mi>p</mi> <mi>e</mi> <mi>r</mi> <mi>t</mi> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>D</mi> <mrow> <mi>P</mi> <mi>o</mi> <mi>s</mi> <mi>s</mi> <mi>i</mi> <mi>b</mi> <mi>l</mi> <mi>e</mi> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>D</mi> <mrow> <mi>C</mi> <mi>o</mi> <mi>m</mi> <mi>p</mi> <mi>E</mi> <mi>x</mi> </mrow> </msub> </semantics></math>. The best regression prediction is by RFR on <math display="inline"><semantics> <msub> <mi>D</mi> <mrow> <mi>C</mi> <mi>o</mi> <mi>m</mi> <mi>p</mi> <mi>E</mi> <mi>x</mi> </mrow> </msub> </semantics></math>.</p> Full article ">Figure 10
<p>Benchmark LR Model coefficients for the merged data from the year 2012.</p> Full article ">Figure 11
<p>Detailed coefficients of base learner based on computational expert features.</p> Full article ">
21 pages, 4702 KiB  
Article
A Four-Variable Shear Deformation Theory for the Static Analysis of FG Sandwich Plates with Different Porosity Models
by Rabab A. Alghanmi and Rawan H. Aljaghthami
Math. Comput. Appl. 2024, 29(2), 20; https://doi.org/10.3390/mca29020020 - 8 Mar 2024
Cited by 3 | Viewed by 1668
Abstract
This study is centered on examining the static bending behavior of sandwich plates featuring functionally graded materials, specifically addressing distinct representations of porosity distribution across their thickness. The composition of the sandwich plate involves a ceramic core and two face sheets with functionally [...] Read more.
This study is centered on examining the static bending behavior of sandwich plates featuring functionally graded materials, specifically addressing distinct representations of porosity distribution across their thickness. The composition of the sandwich plate involves a ceramic core and two face sheets with functionally graded properties. Mechanical loads with a sinusoidal distribution are applied to the sandwich plate, and a four-variable shear deformation theory is employed to establish the displacement field. Notably, this theory involves only four unknowns, distinguishing it from alternative shear deformation theories. Equilibrium equations are derived using the virtual work concept, and Navier’s method is applied to obtain the solution. The study addresses the impact of varying porosities, inhomogeneity parameters, aspect ratios, and side-to-thickness ratios on the static bending behavior of the sandwich plates. The influence of various porosities, inhomogeneity parameter, aspect ratio, and side-to-thickness ratio of the sandwich plates are explored and compared in the context of static bending behavior. The three porosity distributions are compared in terms of their influence on the bending behavior of the sandwich plate. The findings indicate that a higher porosity causes larger deflections and Model A has the highest central deflection. Adopting the four-variable shear deformation theory demonstrated its validity since the results were similar to those obtained in the literature. Several important findings have been found, which could be useful in the construction and application of FG sandwich structures. Examples of comparison will be discussed to support the existing theory’s accuracy. Further findings are presented to serve as benchmarks for comparison. Full article
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<p>Geometry and dimensions of the sandwich plate with porosities.</p> Full article ">Figure 2
<p>Variation in Young’s modulus of porous FG sandwich plate for different values of <math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>α</mi> <mo>=</mo> <mn>0.1</mn> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Variation in Young’s modulus of porous FG sandwich plate for different porosity factor <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The distribution of nondimensional center deflection and stresses in a square FG sandwich plate (Model A) for different porosity factor <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> = 2). (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="false"> <mrow> <mi>w</mi> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>The distribution of nondimensional center deflection and stresses in a square FG sandwich plate (Model B) for different porosity factor <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> = 2). (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="false"> <mrow> <mi>w</mi> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 5 Cont.
<p>The distribution of nondimensional center deflection and stresses in a square FG sandwich plate (Model B) for different porosity factor <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> = 2). (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="false"> <mrow> <mi>w</mi> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>The distribution of nondimensional center deflection and stresses in a square FG sandwich plate (Model C) for different porosity factor <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> = 2). (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="false"> <mrow> <mi>w</mi> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>The distribution of nondimensional center deflection in a square FG sandwich plate (<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> = 2) (Model A) as a function of the following: (<b>a</b>) aspect ratio <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>b</mi> </mrow> </semantics></math>; (<b>b</b>) side-to-thickness ratio <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>The distribution of nondimensional center deflection in a square FG sandwich plate (Model B) as a function of the following: (<b>a</b>) aspect ratio <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>b</mi> </mrow> </semantics></math>; (<b>b</b>) side-to-thickness ratio <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>The distribution of nondimensional center deflection in a square FG sandwich plate (<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> = 2) (Model C) as a function of the following: (<b>a</b>) aspect ratio <span class="html-italic">a</span>/<span class="html-italic">b</span>; (<b>b</b>) side-to-thickness ratio <span class="html-italic">a</span>/<span class="html-italic">h</span>.</p> Full article ">Figure 10
<p>The distribution of nondimensional center deflection and stresses in a square FG sandwich plate for different porosity models (<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> = 2) (<math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> = 0.25). (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="false"> <mrow> <mi>w</mi> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>The distribution of nondimensional center deflection in a square FG sandwich plate for different porosity models as a function of the following: (<b>a</b>) aspect ratio <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>b</mi> </mrow> </semantics></math>; (<b>b</b>) side-to-thickness ratio <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>h</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.25</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>).</p> Full article ">Figure 12
<p>The distribution of nondimensional center deflection and stresses in a square FG sandwich plate with ceramic core (Model B) for different values of <math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> = 0.1). (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="false"> <mrow> <mi>w</mi> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 12 Cont.
<p>The distribution of nondimensional center deflection and stresses in a square FG sandwich plate with ceramic core (Model B) for different values of <math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> = 0.1). (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="false"> <mrow> <mi>w</mi> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 13
<p>The distribution of nondimensional center deflection and stresses in a square FG sandwich plate with metal core (Model B) for different porosity factor <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> = 2). (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="false"> <mrow> <mi>w</mi> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 13 Cont.
<p>The distribution of nondimensional center deflection and stresses in a square FG sandwich plate with metal core (Model B) for different porosity factor <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math> = 2). (<b>a</b>) <math display="inline"><semantics> <mrow> <mover accent="false"> <mrow> <mi>w</mi> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>σ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>τ</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">
25 pages, 5302 KiB  
Article
SSA-Deep Learning Forecasting Methodology with SMA and KF Filters and Residual Analysis
by Juan Frausto-Solís, José Christian de Jesús Galicia-González, Juan Javier González-Barbosa, Guadalupe Castilla-Valdez and Juan Paulo Sánchez-Hernández
Math. Comput. Appl. 2024, 29(2), 19; https://doi.org/10.3390/mca29020019 - 5 Mar 2024
Viewed by 2250
Abstract
Accurate forecasting remains a challenge, even with advanced techniques like deep learning (DL), ARIMA, and Holt–Winters (H&W), particularly for chaotic phenomena such as those observed in several areas, such as COVID-19, energy, and financial time series. Addressing this, we introduce a Forecasting Method [...] Read more.
Accurate forecasting remains a challenge, even with advanced techniques like deep learning (DL), ARIMA, and Holt–Winters (H&W), particularly for chaotic phenomena such as those observed in several areas, such as COVID-19, energy, and financial time series. Addressing this, we introduce a Forecasting Method with Filters and Residual Analysis (FMFRA), a hybrid methodology specifically applied to datasets of COVID-19 time series, which we selected for their complexity and exemplification of current forecasting challenges. FMFFRA consists of the following two approaches: FMFRA-DL, employing deep learning, and FMFRA-SSA, using singular spectrum analysis. This proposed method applies the following three phases: filtering, forecasting, and residual analysis. Initially, each time series is split into filtered and residual components. The second phase involves a simple fine-tuning for the filtered time series, while the third phase refines the forecasts and mitigates noise. FMFRA-DL is adept at forecasting complex series by distinguishing primary trends from insufficient relevant information. FMFRA-SSA is effective in data-scarce scenarios, enhancing forecasts through automated parameter search and residual analysis. Chosen for their geographical and substantial populations and chaotic dynamics, time series for Mexico, the United States, Colombia, and Brazil permitted a comparative perspective. FMFRA demonstrates its efficacy by improving the common forecasting performance measures of MAPE by 22.91%, DA by 13.19%, and RMSE by 25.24% compared to the second-best method, showcasing its potential for providing essential insights into various rapidly evolving domains. Full article
(This article belongs to the Topic Mathematical Modeling)
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<p>Kalman filter estimation: (<b>a</b>) Block algebra of the Plant, (<b>b</b>) block algebra of the Kalman filter.</p> Full article ">Figure 2
<p>FMFRA method.</p> Full article ">Figure 3
<p>Training of FMFRA-DL and FMFRA-SSA to select the best model in the validation set.</p> Full article ">Figure 4
<p>Model 1: Block diagram of the FMFRA-DL method.</p> Full article ">Figure 5
<p>Selection of the best validation forecast between three executions.</p> Full article ">Figure 6
<p>Model 2: Block diagram of the FMFRA-SSA method.</p> Full article ">Figure 7
<p>Training and validation matrix in linear regression format.</p> Full article ">Figure 8
<p>Comparative analysis of population distribution and COVID-19 case frequency in selected American countries. (<b>a</b>) Bar graph population; (<b>b</b>) New cases density distribution.</p> Full article ">Figure 9
<p>Filtered time series of new cases of COVID-19 using SMA in: (<b>a</b>) Mexico, (<b>b</b>) Colombia, (<b>c</b>) USA, and (<b>d</b>) Brazil.</p> Full article ">Figure 10
<p>Update of the uncertainty of the state prediction due to a new measurement of the output.</p> Full article ">Figure 11
<p>Kalman filter for new cases of COVID-19: (<b>a</b>) Mexico, (<b>b</b>) Colombia, (<b>c</b>) USA, and (<b>d</b>) Brazil.</p> Full article ">Figure 12
<p>Comparison of the forecast performance between <math display="inline"><semantics> <mrow> <mi>m</mi> </mrow> </semantics></math> executions: (<b>a</b>) validation set, and (<b>b</b>) test set.</p> Full article ">Figure 13
<p>Standard deviation of the forecast horizon in RMSE for <span class="html-italic">m</span> = 1, 5, 10, 15, 20, and 25. (<b>a</b>) Variability of the forecasting with <span class="html-italic">m</span> = 1, 10, and 20 in green along 21 days; (<b>b</b>) Standard Deviation along the forecasting.</p> Full article ">
27 pages, 1423 KiB  
Article
Energy-and-Blocking-Aware Routing and Device Assignment in Software-Defined Networking—A MILP and Genetic Algorithm Approach
by Gerardo J. Riveros-Rojas, Pedro P. Cespedes-Sanchez, Diego P. Pinto-Roa and Horacio Legal-Ayala
Math. Comput. Appl. 2024, 29(2), 18; https://doi.org/10.3390/mca29020018 - 4 Mar 2024
Cited by 1 | Viewed by 1533
Abstract
Internet energy consumption has increased rapidly, and energy conservation has become a significant issue that requires focused research efforts. The most promising solution is to identify the minimum power subsets within the network and shut down unnecessary network devices and links to satisfy [...] Read more.
Internet energy consumption has increased rapidly, and energy conservation has become a significant issue that requires focused research efforts. The most promising solution is to identify the minimum power subsets within the network and shut down unnecessary network devices and links to satisfy traffic loads. Due to their distributed network control, implementing a centralized and coordinated strategy in traditional networks is challenging. Software-Defined Networking (SDN) is an emerging technology with dynamic, manageable, cost-effective, and adaptable solutions. SDN decouples network control and forwarding functions, allowing network control to be directly programmable, centralizing control with a global network view to manage power states. Nevertheless, it is crucial to develop efficient algorithms that leverage the centralized control of SDN to achieve maximum energy savings and consider peak traffic times. Traffic demand usually cannot be satisfied, even when all network devices are active. This work jointly addresses the routing of traffic flows and the assignment of SDN devices to these flows, called the Routing and Device Assignment (RDA) problem. It simultaneously seeks to minimize the network’s energy consumption and blocked traffic flows. For this approach, we develop an exact solution based on Mixed-Integer Linear Programming (MILP) as well as a metaheuristic based on a Genetic Algorithm (GA) that seeks to optimize both criteria by routing flows efficiently and suspending devices not used by the flows. Conducted simulations on traffic environment scenarios show up to 34% savings in overall energy consumption for the MILP and 33% savings achieved by the GA. These values are better than those obtained using competitive state-of-the-art strategies. Full article
(This article belongs to the Section Engineering)
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<p>Energy use of communication technologies relative to global energy consumption [<a href="#B3-mca-29-00018" class="html-bibr">3</a>].</p> Full article ">Figure 2
<p>Software-Defined Networking architecture [<a href="#B11-mca-29-00018" class="html-bibr">11</a>].</p> Full article ">Figure 3
<p>An overview of the SDN management layer [<a href="#B12-mca-29-00018" class="html-bibr">12</a>].</p> Full article ">Figure 4
<p>SDN switch components [<a href="#B3-mca-29-00018" class="html-bibr">3</a>].</p> Full article ">Figure 5
<p>Expanded diagram of network components.</p> Full article ">Figure 6
<p>Chromosome structure.</p> Full article ">Figure 7
<p>PMX-based crossover of two parents.</p> Full article ">Figure 8
<p>SNDlib topologies [<a href="#B65-mca-29-00018" class="html-bibr">65</a>].</p> Full article ">Figure 9
<p>Atlanta topology comparison with static traffic [<a href="#B65-mca-29-00018" class="html-bibr">65</a>].</p> Full article ">Figure 9 Cont.
<p>Atlanta topology comparison with static traffic [<a href="#B65-mca-29-00018" class="html-bibr">65</a>].</p> Full article ">Figure 10
<p>Abilene topology comparison with static traffic [<a href="#B65-mca-29-00018" class="html-bibr">65</a>].</p> Full article ">Figure 11
<p>Comparison of incremental traffic without re-routing in Atlanta topology for RDA-MILP and RDA-GA.</p> Full article ">Figure 11 Cont.
<p>Comparison of incremental traffic without re-routing in Atlanta topology for RDA-MILP and RDA-GA.</p> Full article ">Figure 12
<p>Comparison of incremental traffic without re-routing in Abilene topology for RDA-MILP and RDA-GA.</p> Full article ">Figure 13
<p>Dynamic traffic comparison in Atlanta topology.</p> Full article ">Figure 14
<p>Comparison of dynamic traffic in Abilene topology.</p> Full article ">Figure 15
<p>Trade-off between blockage rate vs energy saving.</p> Full article ">
23 pages, 392 KiB  
Article
An Iterative Method for Computing π by Argument Reduction of the Tangent Function
by Sanjar M. Abrarov, Rehan Siddiqui, Rajinder Kumar Jagpal and Brendan M. Quine
Math. Comput. Appl. 2024, 29(2), 17; https://doi.org/10.3390/mca29020017 - 25 Feb 2024
Cited by 1 | Viewed by 1620
Abstract
In this work, we develop a new iterative method for computing the digits of π by argument reduction of the tangent function. This method combines a modified version of the iterative formula for π with squared convergence that we proposed in a previous [...] Read more.
In this work, we develop a new iterative method for computing the digits of π by argument reduction of the tangent function. This method combines a modified version of the iterative formula for π with squared convergence that we proposed in a previous work and a leading arctangent term from the Machin-like formula. The computational test we performed shows that algorithmic implementation can provide more than 17 digits of π per increment. Mathematica codes, showing the convergence rate for computing the digits of π, are presented. Full article
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