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Computation
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10th Anniversary of Computation—Computational Engineering
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10th Anniversary of Computation—Computational Engineering

A special issue of Computation (ISSN 2079-3197). This special issue belongs to the section "Computational Engineering".

Deadline for manuscript submissions: closed (31 March 2024) | Viewed by 85797

Special Issue Editors

Prof. Dr. Yudong Zhang

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Guest Editor
Informatics Building School of Informatics, University of Leicester, Leicester LE1 7RH, UK
Interests: explainable deep learning; medical image analysis; pattern recognition and medical sensors; artificial intelligence; intelligent computing
Special Issues, Collections and Topics in MDPI journals
Dr. Francesco Cauteruccio

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Guest Editor
Department of Information Engineering, Polytechnic University of Marche, 60121 Ancona, Italy
Interests: social and complex network analysis; Internet of Things; logic programming and methods for coupling inductive and deductive reasoning; advanced algorithms for sequences comparison; bioinformatics and medical informatics applications; data mining and data science
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Published for the first time in 2013, Computation will celebrate its 10th anniversary this year. In celebration of it, we propose a Special Issue that aims to showcase the cutting-edge research and advancements in the field of Computational Engineering. Over the past decade, the journal has been at the forefront of promoting innovative approaches in computational methods, simulations, and their applications in engineering. This Special Issue will provide a valuable opportunity to highlight the most significant contributions made by researchers in this dynamic and rapidly evolving discipline.

The Special Issue will focus on original research articles, review papers, and case studies that encompass a wide range of topics related to computational engineering. We invite contributions that emphasize novel numerical methods, optimization techniques, data-driven approaches, and the integration of artificial intelligence and machine learning in engineering simulations. Moreover, we encourage submissions that explore the application of computational methods in diverse fields, including, but not limited to, fluid dynamics, structural analysis, materials science, renewable energy systems, and biomedical engineering.

Prof. Dr. Yudong Zhang
Dr. Francesco Cauteruccio
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Computation is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • computational engineering
  • big data
  • data analysis
  • complex engineering phenomena
  • Optimization
  • computational design
  • multiphysics modeling

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Published Papers (40 papers)

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Research

18 pages, 6823 KiB  
Article
Description of Mesoscale Static and Fatigue Analysis of 2D Woven Roving Plates with Convex Holes Subjected to Axial Tension
by Aleksander Muc
Computation 2024, 12(6), 123; https://doi.org/10.3390/computation12060123 - 13 Jun 2024
Viewed by 838
Abstract
The static and fatigue analysis of plates made of 2D woven roving composites with holes is conducted. The parametrization of convex holes is proposed. The experimental results of the specimens without holes and with different shapes of notches are discussed. The experiments and [...] Read more.
The static and fatigue analysis of plates made of 2D woven roving composites with holes is conducted. The parametrization of convex holes is proposed. The experimental results of the specimens without holes and with different shapes of notches are discussed. The experiments and the appropriate procedures are carried out with the aid of ASTM codes. The fatigue behavior is considered with the use of the low cycle fatigue method. The analysis is supplemented by numerical finite element modeling. The present work is an extension of the results discussed in the literature. The damage of plates with holes subjected to tension always occurs at the tip of the holes, i.e., (x = a, b = 0), both for static and fatigue failure. The originality and the novelty of this approach are described by the failure’s dependence on two parameters: n and the ratio of the a/b ratio characterizing the hole geometry. The fuzzy approach is employed to reduce the amount of experimental data. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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Figure 1
<p>Possible approaches in the analysis of composite constructions.</p> Full article ">Figure 2
<p>Commonly 2-D woven roving composite architecture. (<b>a</b>) plain weave; (<b>b</b>) twill weave; (<b>c</b>) satin weave.</p> Full article ">Figure 3
<p>The geometry of the specimens subjected to the uniaxial tension—the lower part of the figure represents the location of the drilled hole.</p> Full article ">Figure 4
<p>Textile 2D woven fabric (plain weave). (<b>a</b>) aramid plain weave (t<sub>ind</sub> = 0.08 mm); (<b>b</b>) glass plain weave (t<sub>ind</sub> = 0.2 mm); (<b>c</b>) a schematic view of plain weave theoretical homogenization.</p> Full article ">Figure 5
<p>ε-N<sub>f</sub> curve—elastic and plastic strain curves.</p> Full article ">Figure 6
<p>Stress–strain curves for LCF.</p> Full article ">Figure 7
<p>Strain life ε-N curve.</p> Full article ">Figure 8
<p>Fatigue tensile load distributions.</p> Full article ">Figure 9
<p>Shape of the supercircle for different values of n (the constant area).</p> Full article ">Figure 10
<p>Static strength of rectangular specimens made of woven roving composites (plain 2D glass) for different warp orientations.</p> Full article ">Figure 11
<p>Boundary conditions for specimens (<a href="#computation-12-00123-f003" class="html-fig">Figure 3</a>) with and without holes—different colors of lines correspond to different forms of boundary conditions.</p> Full article ">Figure 12
<p>Failure modes of 2D woven roving composites subjected to tension. (<b>a</b>) Aramid; (<b>b</b>) Glass.</p> Full article ">Figure 13
<p>Distributions of the stresses (the vertical axes warp, the horizontal axes weft—the width)—the force 11.25 kN.</p> Full article ">Figure 14
<p>Final failure of specimens subjected to uniaxial tensile load—woven roving glass.</p> Full article ">Figure 15
<p>The accuracy of computations for quadrilateral mesh for ellipsoids—static stress analysis.</p> Full article ">Figure 16
<p>Distributions of dimensionless the Huber-Mises-Hencky stresses for n = 2 and the constant area A<sub>c.</sub> (<b>a</b>) a vertical ellipse; (<b>b</b>) a circle; (<b>c</b>) a horizontal ellipse.</p> Full article ">Figure 16 Cont.
<p>Distributions of dimensionless the Huber-Mises-Hencky stresses for n = 2 and the constant area A<sub>c.</sub> (<b>a</b>) a vertical ellipse; (<b>b</b>) a circle; (<b>c</b>) a horizontal ellipse.</p> Full article ">Figure 17
<p>Variations in the stress concentration around convex hole for constant area A<sub>c</sub>.</p> Full article ">Figure 18
<p>Tensile strain distributions around elliptical holes—glass 2D plain weave. (<b>a</b>) b/a = 0.2 (the minimal blue; the maximum is white); (<b>b</b>) b/a = 0.5 (the minimal dark green; the maximal light green).</p> Full article ">Figure 19
<p>Fatigue failure modes of stretched plates made of woven roving glass/epoxy—circular hole.</p> Full article ">Figure 20
<p>The degradation of the stiffness for 2D woven roving composites (M—the average value, R—the highest value treated as the upper bound, L—the lowest value treated as the lower bound).</p> Full article ">Figure 21
<p>The accuracy of computations for triangular mesh for circular holes—fatigue (LCS) analysis.</p> Full article ">Figure 22
<p>Fatigue crack initiation life contours (the logarithmic scale).</p> Full article ">Figure 23
<p>Variations in the critical number of cycles N<sub>f</sub> around convex hole for constant area A<sub>c</sub>.</p> Full article ">
17 pages, 7133 KiB  
Article
Deep-Reinforcement-Learning-Based Motion Planning for a Wide Range of Robotic Structures
by Roman Parák, Jakub Kůdela, Radomil Matoušek and Martin Juříček
Computation 2024, 12(6), 116; https://doi.org/10.3390/computation12060116 - 5 Jun 2024
Viewed by 2524
Abstract
The use of robot manipulators in engineering applications and scientific research has significantly increased in recent years. This can be attributed to the rise of technologies such as autonomous robotics and physics-based simulation, along with the utilization of artificial intelligence techniques. The use [...] Read more.
The use of robot manipulators in engineering applications and scientific research has significantly increased in recent years. This can be attributed to the rise of technologies such as autonomous robotics and physics-based simulation, along with the utilization of artificial intelligence techniques. The use of these technologies may be limited due to a focus on a specific type of robotic manipulator and a particular solved task, which can hinder modularity and reproducibility in future expansions. This paper presents a method for planning motion across a wide range of robotic structures using deep reinforcement learning (DRL) algorithms to solve the problem of reaching a static or random target within a pre-defined configuration space. The paper addresses the challenge of motion planning in environments under a variety of conditions, including environments with and without the presence of collision objects. It highlights the versatility and potential for future expansion through the integration of OpenAI Gym and the PyBullet physics-based simulator. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>A schematic representation of a motion planning problem in two-dimensional space, illustrating an n-link robotic manipulator <math display="inline"><semantics> <mi mathvariant="script">R</mi> </semantics></math> within a collision-free configuration space <math display="inline"><semantics> <msub> <mi mathvariant="script">C</mi> <mi>free</mi> </msub> </semantics></math> with obstacles denoted as <math display="inline"><semantics> <mi mathvariant="script">O</mi> </semantics></math>.</p> Full article ">Figure 2
<p>An illustration of the robotic structures that are part of the Industry 4.0 Cell (I4C).</p> Full article ">Figure 3
<p>The interaction between an agent and the environment in a Markov Decision Process (MDP), adapted from [<a href="#B14-computation-12-00116" class="html-bibr">14</a>].</p> Full article ">Figure 4
<p>An overview of the actor–critic architecture. The Temporal Difference (TD) error <math display="inline"><semantics> <msub> <mi>δ</mi> <mi>t</mi> </msub> </semantics></math> is utilized to adjust both the critic’s action value function <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> and the actor’s policy <math display="inline"><semantics> <mrow> <mi>π</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∣</mo> <mi>s</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>, which is parameterized by <math display="inline"><semantics> <mi>θ</mi> </semantics></math> [<a href="#B14-computation-12-00116" class="html-bibr">14</a>,<a href="#B16-computation-12-00116" class="html-bibr">16</a>].</p> Full article ">Figure 5
<p>An illustration of both types of environments, <math display="inline"><semantics> <msub> <mi mathvariant="script">E</mi> <mn>1</mn> </msub> </semantics></math> (<b>left</b>) and <math display="inline"><semantics> <msub> <mi mathvariant="script">E</mi> <mn>2</mn> </msub> </semantics></math> (<b>right</b>), that were used in an experiment focused on reaching the target in a pre-defined configuration space <math display="inline"><semantics> <msub> <mi mathvariant="script">C</mi> <mi>free</mi> </msub> </semantics></math>. The yellow wireframe determines a pre-defined configuration space <math display="inline"><semantics> <msub> <mi mathvariant="script">C</mi> <mi>free</mi> </msub> </semantics></math>, while the green wireframe delineates the area where the target was randomly generated. The red sphere within the environment <math display="inline"><semantics> <msub> <mi mathvariant="script">E</mi> <mn>2</mn> </msub> </semantics></math> represents an obstacle approximated with Axis-Aligned Bounding Box (AABB), denoted as <math display="inline"><semantics> <msub> <mi mathvariant="script">O</mi> <mrow> <mi>A</mi> <mi>A</mi> <mi>B</mi> <mi>B</mi> </mrow> </msub> </semantics></math>.</p> Full article ">Figure 6
<p>The training process shows success rates within the environment type <math display="inline"><semantics> <msub> <mi mathvariant="script">E</mi> <mn>1</mn> </msub> </semantics></math> for the DDPG, SAC, and TD3 algorithms (<b>left</b>) and an extension with HER (<b>right</b>).</p> Full article ">Figure 7
<p>The training process shows success rates within the environment type <math display="inline"><semantics> <msub> <mi mathvariant="script">E</mi> <mn>2</mn> </msub> </semantics></math> for the DDPG, SAC, and TD3 algorithms (<b>left</b>) and an extension with HER (<b>right</b>).</p> Full article ">Figure 8
<p>An illustration of a wide range of robotic structures within the <math display="inline"><semantics> <msub> <mi mathvariant="script">E</mi> <mn>1</mn> </msub> </semantics></math> environment, simulated using a PyBullet physics-based simulator. These structures were used in an experiment that focused on reaching the target in a pre-defined configuration space.</p> Full article ">Figure 8 Cont.
<p>An illustration of a wide range of robotic structures within the <math display="inline"><semantics> <msub> <mi mathvariant="script">E</mi> <mn>1</mn> </msub> </semantics></math> environment, simulated using a PyBullet physics-based simulator. These structures were used in an experiment that focused on reaching the target in a pre-defined configuration space.</p> Full article ">Figure 9
<p>An illustration of a wide range of robotic structures within the <math display="inline"><semantics> <msub> <mi mathvariant="script">E</mi> <mn>2</mn> </msub> </semantics></math> environment, simulated using PyBullet physics-based simulator. These structures were used in an experiment that focused on reaching the target in a pre-defined configuration space.</p> Full article ">
16 pages, 10027 KiB  
Article
Minimizing the Number of Distrustful Nodes on the Path of IP Packet Transmission
by Kvitoslava Obelovska, Oleksandr Tkachuk and Yaromyr Snaichuk
Computation 2024, 12(5), 91; https://doi.org/10.3390/computation12050091 - 3 May 2024
Cited by 1 | Viewed by 1007
Abstract
One of the important directions for improving modern Wide Area Networks is efficient and secure packet routing. Efficient routing is often based on using the shortest paths, while ensuring security involves preventing the possibility of packet interception. The work is devoted to improving [...] Read more.
One of the important directions for improving modern Wide Area Networks is efficient and secure packet routing. Efficient routing is often based on using the shortest paths, while ensuring security involves preventing the possibility of packet interception. The work is devoted to improving the security of data transmission in IP networks. A new approach is proposed to minimize the number of distrustful nodes on the path of IP packet transmission. By a distrustful node, we mean a node that works correctly in terms of hardware and software and fully implements its data transport functions, but from the point of view of its organizational subordination, we are not sure that the node will not violate security rules to prevent unauthorized access and interception of data. A distrustful node can be either a transit or an end node. To implement this approach, we modified Dijkstra’s shortest path tree construction algorithm. The modified algorithm ensures that we obtain a path that will pass only through trustful nodes, if such a path exists. If there is no such path, the path will have the minimum possible number of distrustful intermediate nodes. The number of intermediate nodes in the path was used as a metric to obtain the shortest path trees. Routing tables of routers, built on the basis of trees obtained using a modified algorithm, provide increased security of data transmission, minimizing the use of distrustful nodes. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Flowchart diagram of modified algorithm.</p> Full article ">Figure 2
<p>Network topology of a small network.</p> Full article ">Figure 3
<p>The shortest path tree for node A, according to classical Dijkstra’s algorithm.</p> Full article ">Figure 4
<p>The shortest path tree for node A, according to the modified Dijkstra algorithm.</p> Full article ">Figure 5
<p>Fragments of routing tables: on the left for the classic Dijkstra algorithm, on the right for the modified one, the changed paths are circled in red.</p> Full article ">Figure 6
<p>Network topology with several distrustful nodes.</p> Full article ">Figure 7
<p>The shortest path tree for the network with multiple distrustful nodes, according to the classical Dijkstra algorithm.</p> Full article ">Figure 8
<p>The shortest path tree for the network with multiple distrustful nodes, according to the modified Dijkstra algorithm.</p> Full article ">Figure 9
<p>Fragments of routing tables for a network with three distrustful nodes: on the left for the classic Dijkstra algorithm, on the right for the modified one, the changed path is circled in red.</p> Full article ">Figure 10
<p>Network topology with 16 nodes and 2 distrustful nodes.</p> Full article ">Figure 11
<p>The shortest path tree for the (<b>a</b>) classical Dijkstra algorithm and (<b>b</b>) modified Dijkstra algorithm.</p> Full article ">Figure 12
<p>Fragments of routing tables for the network of <a href="#computation-12-00091-f009" class="html-fig">Figure 9</a>, according to the (<b>a</b>) classical Dijkstra algorithm and (<b>b</b>) modified Dijkstra algorithm.</p> Full article ">Figure 13
<p>Network topology with 20 nodes and 3 distrustful nodes.</p> Full article ">Figure 14
<p>The shortest path tree for the (<b>a</b>) classical Dijkstra algorithm and (<b>b</b>) modified Dijkstra algorithm.</p> Full article ">Figure 15
<p>The topology of a large network.</p> Full article ">Figure 16
<p>The shortest path tree, according to the classical Dijkstra algorithm.</p> Full article ">Figure 17
<p>The shortest path tree, according to the modified Dijkstra algorithm.</p> Full article ">
20 pages, 1879 KiB  
Article
A Weighted and Epsilon-Constraint Biased-Randomized Algorithm for the Biobjective TOP with Prioritized Nodes
by Lucia Agud-Albesa, Neus Garrido, Angel A. Juan, Almudena Llorens and Sandra Oltra-Crespo
Computation 2024, 12(4), 84; https://doi.org/10.3390/computation12040084 - 20 Apr 2024
Viewed by 1038
Abstract
This paper addresses a multiobjective version of the Team Orienteering Problem (TOP). The TOP focuses on selecting a subset of customers for maximum rewards while considering time and fleet size constraints. This study extends the TOP by considering two objectives: maximizing total rewards [...] Read more.
This paper addresses a multiobjective version of the Team Orienteering Problem (TOP). The TOP focuses on selecting a subset of customers for maximum rewards while considering time and fleet size constraints. This study extends the TOP by considering two objectives: maximizing total rewards from customer visits and maximizing visits to prioritized nodes. The MultiObjective TOP (MO-TOP) is formulated mathematically to concurrently tackle these objectives. A multistart biased-randomized algorithm is proposed to solve MO-TOP, integrating exploration and exploitation techniques. The algorithm employs a constructive heuristic defining biefficiency to select edges for routing plans. Through iterative exploration from various starting points, the algorithm converges to high-quality solutions. The Pareto frontier for the MO-TOP is generated using the weighted method, epsilon-constraint method, and Epsilon-Modified Method. Computational experiments validate the proposed approach’s effectiveness, illustrating its ability to generate diverse and high-quality solutions on the Pareto frontier. The algorithms demonstrate the ability to optimize rewards and prioritize node visits, offering valuable insights for real-world decision making in team orienteering applications. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>A basic schema of the multiobjective TOP considered in this paper.</p> Full article ">Figure 2
<p>Graph for the instance p5.4.r with the EMM.</p> Full article ">Figure 3
<p>Graph for the instance p5.4.r with the POWAM.</p> Full article ">Figure 4
<p>Pareto frontier for the MO-TOP p44o.</p> Full article ">Figure 5
<p>Pareto frontier for the MO-TOP p44r.</p> Full article ">Figure 6
<p>Pareto frontier for the MO-TOP p54q.</p> Full article ">Figure 7
<p>Pareto frontier for the MO-TOP p54r.</p> Full article ">Figure 8
<p>Pareto frontier for the MO-TOP p74q.</p> Full article ">Figure 9
<p>Pareto frontier for the MO-TOP p74r.</p> Full article ">Figure 10
<p>Solutions for the MO-TOP p44o using all the described methods for rewards.</p> Full article ">Figure 11
<p>Solutions for the MO-TOP p44o using all the described methods for priority nodes visited.</p> Full article ">
10 pages, 1126 KiB  
Article
Application of Machine Learning to Predict Blockage in Multiphase Flow
by Nazerke Saparbayeva, Boris V. Balakin, Pavel G. Struchalin, Talal Rahman and Sergey Alyaev
Computation 2024, 12(4), 67; https://doi.org/10.3390/computation12040067 - 31 Mar 2024
Viewed by 1656
Abstract
This study presents a machine learning-based approach to predict blockage in multiphase flow with cohesive particles. The aim is to predict blockage based on parameters like Reynolds and capillary numbers using a random forest classifier trained on experimental and simulation data. Experimental observations [...] Read more.
This study presents a machine learning-based approach to predict blockage in multiphase flow with cohesive particles. The aim is to predict blockage based on parameters like Reynolds and capillary numbers using a random forest classifier trained on experimental and simulation data. Experimental observations come from a lab-scale flow loop with ice slurry in the decane. The plugging simulation is based on coupled Computational Fluid Dynamics with Discrete Element Method (CFD-DEM). The resulting classifier demonstrated high accuracy, validated by precision, recall, and F1-score metrics, providing precise blockage prediction under specific flow conditions. Additionally, sensitivity analyses highlighted the model’s adaptability to cohesion variations. Equipped with the trained classifier, we generated a detailed machine-learning-based flow map and compared it with earlier literature, simulations, and experimental data results. This graphical representation clarifies the blockage boundaries under given conditions. The methodology’s success demonstrates the potential for advanced predictive modelling in diverse flow systems, contributing to improved blockage prediction and prevention. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Hydraulic diagram of the experimental flow loop (<b>A</b>) with the orifice indicated by an orange arrow and photo of the central part of the flow loop (<b>B</b>).</p> Full article ">Figure 2
<p>Geometry and boundary conditions of the model (<b>left</b>) and comparison between experimental data and CFD-DEM model: average flow velocity over time with different cohesion-to-adhesion ratios <math display="inline"><semantics> <msub> <mi>c</mi> <mi>r</mi> </msub> </semantics></math> = <math display="inline"><semantics> <msub> <mi>γ</mi> <mrow> <mi>w</mi> <mi>a</mi> <mi>l</mi> <mi>l</mi> </mrow> </msub> </semantics></math>/<math display="inline"><semantics> <msub> <mi>γ</mi> <mrow> <mi>i</mi> <mi>c</mi> <mi>e</mi> </mrow> </msub> </semantics></math> in comparison to the findings of Struchalin et al. [<a href="#B13-computation-12-00067" class="html-bibr">13</a>] at Reynolds number Re = 4996 and particle volume fraction <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mi>p</mi> </msub> </semantics></math> = 6.8% (<b>right</b>).</p> Full article ">Figure 3
<p>Schematic description of the developed ML model.</p> Full article ">Figure 4
<p>ML-predicted process conditions resulted with plugging. Intermediate result produced from the original dataset.</p> Full article ">Figure 5
<p>Flow map from the random forest classifier with three different cohesion values. The datapoints in the plot represent experimental [<a href="#B13-computation-12-00067" class="html-bibr">13</a>] and the CFD-DEM cases where the blockage was detected. Experimental points excluded from training are labeled with star-like markers. Flow map from Hirochi et al. [<a href="#B14-computation-12-00067" class="html-bibr">14</a>] shown for comparison. The lines present the boundaries of the plugging regime predicted by the ML-model and shown in Hirochi et al. [<a href="#B14-computation-12-00067" class="html-bibr">14</a>].</p> Full article ">
19 pages, 12184 KiB  
Article
Generalized Approach to Optimal Polylinearization for Smart Sensors and Internet of Things Devices
by Marin B. Marinov and Slav Dimitrov
Computation 2024, 12(4), 63; https://doi.org/10.3390/computation12040063 - 23 Mar 2024
Viewed by 1189
Abstract
This study introduces an innovative numerical approach for polylinear approximation (polylinearization) of non-self-intersecting compact sensor characteristics (transfer functions) specified either pointwise or analytically. The goal is to partition the sensor characteristic optimally, i.e., to select the vertices of the approximating polyline (approximant) along [...] Read more.
This study introduces an innovative numerical approach for polylinear approximation (polylinearization) of non-self-intersecting compact sensor characteristics (transfer functions) specified either pointwise or analytically. The goal is to partition the sensor characteristic optimally, i.e., to select the vertices of the approximating polyline (approximant) along with their positions, on the sensor characteristics so that the distance (i.e., the separation) between the approximant and the characteristic is rendered below a certain problem-specific tolerance. To achieve this goal, two alternative nonlinear optimization problems are solved, which differ in the adopted quantitative measure of the separation between the transfer function and the approximant. In the first problem, which relates to absolutely integrable sensor characteristics (their energy is not necessarily finite, but they can be represented in terms of convergent Fourier series), the polylinearization is constructed by the numerical minimization of the L1-metric (a distance-based separation measure), concerning the number of polyline vertices and their locations. In the second problem, which covers the quadratically integrable sensor characteristics (whose energy is finite, but they do not necessarily admit a representation in terms of convergent Fourier series), the polylinearization is constructed by numerically minimizing the L2-metric (area- or energy-based separation measure) for the same set of optimization variables—the locations and the number of polyline vertices. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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Figure 1
<p>(<b>a</b>) Smooth sensor transfer function; (<b>b</b>) polylinearization of the sensor transfer function and the approximating polyline, together with its vertices shown in red.</p> Full article ">Figure 2
<p>(<b>a</b>) a polyline through <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">c</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mrow> <mi mathvariant="bold-italic">c</mi> </mrow> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">c</mi> </mrow> <mrow> <mi>i</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>;</mo> </mrow> </semantics></math> (<b>b</b>) distance-based remoteness—sensor transfer characteristics and a polyline are as far away from each other as the largest projected distance, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mi>m</mi> <mi>a</mi> <mi>x</mi> <mo stretchy="false">{</mo> <msub> <mrow> <mi>d</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mrow> <mi>d</mi> </mrow> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> </semantics></math>; (<b>c</b>) and area-based remoteness—polyline is as far away from the sensor curve as the area <span style="color:#DA5319">■</span> is close to the area <span style="color:#0072BE">■</span> (overlapped by <span style="color:#DA5319">■</span> and not fully visible).</p> Full article ">Figure 3
<p>Illustrations of the concept of remoteness measure for mesh patch <math display="inline"><semantics> <mrow> <mi mathvariant="script">S</mi> <mo>=</mo> <msup> <mrow> <mi mathvariant="script">S</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>∪</mo> <msup> <mrow> <mi mathvariant="script">S</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </semantics></math>: (<b>a</b>) <math display="inline"><semantics> <mrow> <msup> <mrow> <mi mathvariant="script">R</mi> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>2</mn> </mrow> </mfenced> </mrow> </msup> <mfenced separators="|"> <mrow> <mi mathvariant="script">S</mi> </mrow> </mfenced> </mrow> </semantics></math> equals the largest of the differences in the areas under the curve segments and their approximating linear segments and (<b>b</b>) <math display="inline"><semantics> <mrow> <msup> <mrow> <mi mathvariant="script">R</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="script">S</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> equals the largest of the distances between the curve segments and their approximating linear segments.</p> Full article ">Figure 4
<p>Flow chart of the developed algorithm.</p> Full article ">Figure 5
<p>Polylinearization of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mi>s</mi> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> norm. (<b>a</b>) First partition <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and (<b>b</b>) final partition <math display="inline"><semantics> <mrow> <mtext>  </mtext> <msubsup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> <mrow> <mi mathvariant="normal">*</mi> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Polylinearization of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mi>s</mi> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mo>∞</mo> </mrow> </msup> </mrow> </semantics></math> norm. (<b>a</b>) First partition <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and (<b>b</b>) final partition <math display="inline"><semantics> <mrow> <mtext>  </mtext> <msubsup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> <mrow> <mi mathvariant="normal">*</mi> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Polylinearization of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mi>s</mi> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> norm. (<b>a</b>) First partition <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and (<b>b</b>) final partition <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>50</mn> </mrow> <mrow> <mi mathvariant="normal">*</mi> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Polylinearization of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mi>s</mi> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mo>∞</mo> </mrow> </msup> </mrow> </semantics></math> norm. (<b>a</b>) First partition <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and (<b>b</b>) final partition <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>7</mn> </mrow> <mrow> <mi mathvariant="normal">*</mi> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Polylinearization of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mi>s</mi> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msub> <mfenced separators="|"> <mrow> <mi>s</mi> <mo>−</mo> <mn>100</mn> </mrow> </mfenced> </mrow> </mfenced> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> norm. (<b>a</b>) First partition <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and (<b>b</b>) final partition <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>4</mn> </mrow> <mrow> <mi mathvariant="normal">*</mi> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>Polylinearization of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mi>s</mi> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msub> <mfenced separators="|"> <mrow> <mi>s</mi> <mo>−</mo> <mn>100</mn> </mrow> </mfenced> </mrow> </mfenced> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mo>∞</mo> </mrow> </msup> </mrow> </semantics></math> norm. (<b>a</b>) First partition <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and (<b>b</b>) final partition <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>3</mn> </mrow> <mrow> <mi mathvariant="normal">*</mi> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>Polylinearization of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>5</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>5</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mi>s</mi> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> norm. (<b>a</b>) First partition <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and (<b>b</b>) final partition <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>11</mn> </mrow> <mrow> <mi mathvariant="normal">*</mi> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>Polylinearization of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>5</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>5</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mi>s</mi> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mo>∞</mo> </mrow> </msup> </mrow> </semantics></math> norm. (<b>a</b>) First partition <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and (<b>b</b>) final partition <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>5</mn> </mrow> <mrow> <mi mathvariant="normal">*</mi> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 13
<p>Polylinearization of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mfrac> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>π</mi> <mfenced separators="|"> <mrow> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </mfenced> </mrow> </mfrac> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> norm. (<b>a</b>) First partition <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and (<b>b</b>) final partition <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>9</mn> </mrow> <mrow> <mi mathvariant="normal">*</mi> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 14
<p>Polylinearization of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mfrac> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>π</mi> <mfenced separators="|"> <mrow> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </mfenced> </mrow> </mfrac> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mo>∞</mo> </mrow> </msup> </mrow> </semantics></math> norm. (<b>a</b>) First partition <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and (<b>b</b>) final partition <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>6</mn> </mrow> <mrow> <mi mathvariant="normal">*</mi> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 15
<p>Polylinearization of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mfrac> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> <msup> <mrow> <mfenced separators="|"> <mrow> <mfrac> <mrow> <mi>s</mi> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mrow> <mi mathvariant="normal">exp</mi> </mrow> <mo>⁡</mo> <mrow> <mfenced separators="|"> <mrow> <mo>−</mo> <msup> <mrow> <mfenced separators="|"> <mrow> <mfrac> <mrow> <mi>s</mi> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mrow> </mfenced> </mrow> </mrow> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> norm. (<b>a</b>) First partition <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and (<b>b</b>) final partition <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>6</mn> </mrow> <mrow> <mi mathvariant="normal">*</mi> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">Figure 16
<p>Polylinearization of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mfrac> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> <msup> <mrow> <mfenced separators="|"> <mrow> <mfrac> <mrow> <mi>s</mi> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mrow> <mi mathvariant="normal">exp</mi> </mrow> <mo>⁡</mo> <mrow> <mfenced separators="|"> <mrow> <mo>−</mo> <msup> <mrow> <mfenced separators="|"> <mrow> <mfrac> <mrow> <mi>s</mi> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mfenced> </mrow> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mrow> </mfenced> </mrow> </mrow> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mo>∞</mo> </mrow> </msup> </mrow> </semantics></math> norm. (<b>a</b>) First partition <math display="inline"><semantics> <mrow> <msub> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and (<b>b</b>) final partition <math display="inline"><semantics> <mrow> <msubsup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mn>4</mn> </mrow> <mrow> <mi mathvariant="normal">*</mi> </mrow> </msubsup> </mrow> </semantics></math>.</p> Full article ">
13 pages, 557 KiB  
Article
Exploring Numba and CuPy for GPU-Accelerated Monte Carlo Radiation Transport
by Tair Askar, Argyn Yergaliyev, Bekdaulet Shukirgaliyev and Ernazar Abdikamalov
Computation 2024, 12(3), 61; https://doi.org/10.3390/computation12030061 - 20 Mar 2024
Viewed by 2725
Abstract
This paper examines the performance of two popular GPU programming platforms, Numba and CuPy, for Monte Carlo radiation transport calculations. We conducted tests involving random number generation and one-dimensional Monte Carlo radiation transport in plane-parallel geometry on three GPU cards: NVIDIA Tesla A100, [...] Read more.
This paper examines the performance of two popular GPU programming platforms, Numba and CuPy, for Monte Carlo radiation transport calculations. We conducted tests involving random number generation and one-dimensional Monte Carlo radiation transport in plane-parallel geometry on three GPU cards: NVIDIA Tesla A100, Tesla V100, and GeForce RTX3080. We compared Numba and CuPy to each other and our CUDA C implementation. The results show that CUDA C, as expected, has the fastest performance and highest energy efficiency, while Numba offers comparable performance when data movement is minimal. While CuPy offers ease of implementation, it performs slower for compute-heavy tasks. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Execution time as a function of the number of PRNs/particles for CuPy, Numba, and CUDA C using the A100 GPU card. The left panel represents the PRN generation test problem, while the right panel is for the 1D MCRT test. The solid lines show the total execution time and the dashed lines correspond to the GPU kernel times.</p> Full article ">Figure 2
<p>Execution times of GPU kernels as a function of the number of PRNs/particles for CuPy, Numba, and CUDA C using the A100 GPU card. The left panel represents the PRN generation test problem, while the right panel is for the 1D MCRT test problem. The solid lines correspond to single-precision calculations and the dashed lines show double-precision results.</p> Full article ">Figure 3
<p>Execution time of GPU kernel as a function of the number of particles for Numba and CuPy using three GPU cards for the 1D MCRT test problem. The left panel shows the results for Numba, while the right panel is for CuPy.</p> Full article ">
13 pages, 298 KiB  
Article
Practical Improvement in the Implementation of Two Avalanche Tests to Measure Statistical Independence in Stream Ciphers
by Evaristo José Madarro-Capó, Eziel Christians Ramos Piñón, Guillermo Sosa-Gómez and Omar Rojas
Computation 2024, 12(3), 60; https://doi.org/10.3390/computation12030060 - 19 Mar 2024
Viewed by 1571
Abstract
This study describes the implementation of two algorithms in a parallel environment. These algorithms correspond to two statistical tests based on the bit’s independence criterion and the strict avalanche criterion. They are utilized to measure avalanche properties in stream ciphers. These criteria allow [...] Read more.
This study describes the implementation of two algorithms in a parallel environment. These algorithms correspond to two statistical tests based on the bit’s independence criterion and the strict avalanche criterion. They are utilized to measure avalanche properties in stream ciphers. These criteria allow for the statistical independence between the outputs and the internal state of a bit-level cipher to be determined. Both tests require extensive input parameters to assess the performance of current stream ciphers, leading to longer execution times. The presented implementation significantly reduces the execution time of both tests, making them suitable for evaluating ciphers in practical applications. The evaluation results compare the performance of the RC4 and HC256 stream ciphers in both sequential and parallel environments. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Comparison in time of parallel and sequential execution of SAC and BIC algorithms, for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>256</mn> <mo>,</mo> <mo> </mo> <mi>m</mi> <mo>=</mo> <mn>32</mn> </mrow> </semantics></math>, in RC4 with <span class="html-italic">L</span> in {1000, 4096, 5000, 8192, 10,000, 15,000, 16,384}.</p> Full article ">Figure 2
<p>Comparison in time of parallel and sequential execution of SAC and BIC algorithms, for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>256</mn> <mo>,</mo> <mo> </mo> <mi>m</mi> <mo>=</mo> <mn>32</mn> </mrow> </semantics></math>, in HC256 with <span class="html-italic">L</span> in {1000, 4096, 5000, 8192, 10,000, 15,000, 16,384}.</p> Full article ">Figure 3
<p>Comparison in time of parallel and sequential execution of BIC algorithm, for the same parameters described above in HC256 and RC4.</p> Full article ">Figure 4
<p>Comparison in time of parallel and sequential execution of the SAC algorithm, for the same parameters described above in HC256 and RC4.</p> Full article ">
18 pages, 4576 KiB  
Article
High-Compression Crash Simulations and Tests of PLA Cubes Fabricated Using Additive Manufacturing FDM with a Scaling Strategy
by Andres-Amador Garcia-Granada
Computation 2024, 12(3), 40; https://doi.org/10.3390/computation12030040 - 23 Feb 2024
Cited by 1 | Viewed by 1929
Abstract
Impacts due to drops or crashes between moving vehicles necessitate the search for energy absorption elements to prevent damage to the transported goods or individuals. To ensure safety, a given level of acceptable deceleration is provided. The optimization of deformable parts to absorb [...] Read more.
Impacts due to drops or crashes between moving vehicles necessitate the search for energy absorption elements to prevent damage to the transported goods or individuals. To ensure safety, a given level of acceptable deceleration is provided. The optimization of deformable parts to absorb impact energy is typically conducted through explicit simulations, where kinetic energy is converted into plastic deformation energy. The introduction of additive manufacturing techniques enables this optimization to be conducted with more efficient shapes, previously unachievable with conventional manufacturing methods. This paper presents an initial approach to validating explicit simulations of impacts against solid cubes of varying sizes and fabrication directions. Such cubes were fabricated using PLA, the most used material, and a desktop printer. All simulations could be conducted using a single material law description, employing solid elements with a controlled time step suitable for industrial applications. With this approach, the simulations were capable of predicting deceleration levels across a broad range of impact configurations for solid cubes. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Ultimaker Cura slicing for odd and even layers and an isometric view of all layers.</p> Full article ">Figure 2
<p>Compression testing of cubes of different sizes in different directions for <span class="html-italic">L</span> = 15 and 20 mm.</p> Full article ">Figure 3
<p>Simulation of compression tests of cubes of different sizes in different directions.</p> Full article ">Figure 4
<p>Repeatability of experimental compression tests for <span class="html-italic">L</span> = 10 mm.</p> Full article ">Figure 5
<p>Effect of orientation in experimental compression tests for all sizes.</p> Full article ">Figure 6
<p>Energy density obtained from experimental compression tests for <span class="html-italic">L</span> = 10, 15, 20, and 25 mm. Energy values at 40%, 46%, and 50% are provided for impact estimation.</p> Full article ">Figure 7
<p>Force-versus-displacement curves for experimental tests and compression simulations for <span class="html-italic">L</span> = 10, 15, 20, and 25 mm.</p> Full article ">Figure 8
<p>Stress–strain curve from experiments and compression simulation: (<b>a</b>) full scale and (<b>b</b>) detail for 20–25% strain.</p> Full article ">Figure 9
<p>Acceleration–time curves obtained from the integration of experiments (Test) and impact simulations (SIM).</p> Full article ">Figure 10
<p>Acceleration–time curves from the experiment (Test) compared to those from the simulations using different time steps for the worst correlation case for the smallest <span class="html-italic">L</span>, namely, <span class="html-italic">L</span> = 10 mm.</p> Full article ">Figure 11
<p>Time step used in simulation as a function of simulation time for compression and impact simulation.</p> Full article ">Figure 12
<p>Engineering strain rate and true strain rate during simulations of impact on PLA specimen.</p> Full article ">
13 pages, 864 KiB  
Article
Injury Patterns and Impact on Performance in the NBA League Using Sports Analytics
by Vangelis Sarlis, George Papageorgiou and Christos Tjortjis
Computation 2024, 12(2), 36; https://doi.org/10.3390/computation12020036 - 16 Feb 2024
Cited by 5 | Viewed by 7100
Abstract
This research paper examines Sports Analytics, focusing on injury patterns in the National Basketball Association (NBA) and their impact on players’ performance. It employs a unique dataset to identify common NBA injuries, determine the most affected anatomical areas, and analyze how these injuries [...] Read more.
This research paper examines Sports Analytics, focusing on injury patterns in the National Basketball Association (NBA) and their impact on players’ performance. It employs a unique dataset to identify common NBA injuries, determine the most affected anatomical areas, and analyze how these injuries influence players’ post-recovery performance. This study’s novelty lies in its integrative approach that combines injury data with performance metrics and salary data, providing new insights into the relationship between injuries and economic and on-court performance. It investigates the periodicity and seasonality of injuries, seeking patterns related to time and external factors. Additionally, it examines the effect of specific injuries on players’ per-match analytics and performance, offering perspectives on the implications of injury rehabilitation for player performance. This paper contributes significantly to sports analytics, assisting coaches, sports medicine professionals, and team management in developing injury prevention strategies, optimizing player rotations, and creating targeted rehabilitation plans. Its findings illuminate the interplay between injuries, salaries, and performance in the NBA, aiming to enhance player welfare and the league’s overall competitiveness. With a comprehensive and sophisticated analysis, this research offers unprecedented insights into the dynamics of injuries and their long-term effects on athletes. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Musculoskeletal anatomical sub-areas’ statistical significance and salary correlation.</p> Full article ">Figure 2
<p>Tornado diagram that analyzes the percentage variance in basketball performance analytics in lesser/greater post-injury cases.</p> Full article ">
17 pages, 709 KiB  
Article
Accelerating Multiple Sequence Alignments Using Parallel Computing
by Qanita Bani Baker, Ruba A. Al-Hussien and Mahmoud Al-Ayyoub
Computation 2024, 12(2), 32; https://doi.org/10.3390/computation12020032 - 9 Feb 2024
Cited by 1 | Viewed by 2373
Abstract
Multiple sequence alignment (MSA) stands as a critical tool for understanding the evolutionary and functional relationships among biological sequences. Obtaining an exact solution for MSA, termed exact-MSA, is a significant challenge due to the combinatorial nature of the problem. Using the dynamic [...] Read more.
Multiple sequence alignment (MSA) stands as a critical tool for understanding the evolutionary and functional relationships among biological sequences. Obtaining an exact solution for MSA, termed exact-MSA, is a significant challenge due to the combinatorial nature of the problem. Using the dynamic programming technique to solve MSA is recognized as a highly computationally complex algorithm. To cope with the computational demands of MSA, parallel computing offers the potential for significant speedup in MSA. In this study, we investigated the utilization of parallelization to solve the exact-MSA using three proposed novel approaches. In these approaches, we used multi-threading techniques to improve the performance of the dynamic programming algorithms in solving the exact-MSA. We developed and employed three parallel approaches, named diagonal traversing, blocking, and slicing, to improve MSA performance. The proposed method accelerated the exact-MSA algorithm by around 4×. The suggested approaches could be basic approaches to be combined with many existing techniques. These proposed approaches could serve as foundational elements, offering potential integration with existing techniques for comprehensive MSA enhancement. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Values used to compute cell <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> </semantics></math> in the 2D array created for 2-sequence alignment.</p> Full article ">Figure 2
<p>Possible solutions for three sequences.</p> Full article ">Figure 3
<p>The Dynamic programming for MSA.</p> Full article ">Figure 4
<p>Data dependency between cells.</p> Full article ">Figure 5
<p>Traversing methods used to fill scoring matrix.</p> Full article ">Figure 6
<p>The proposed approach 1 to solve 3D problem.</p> Full article ">Figure 7
<p>The proposed approach 2 to solve 3D problem.</p> Full article ">
21 pages, 8030 KiB  
Article
Numerical Modeling and Analysis of Transient and Three-Dimensional Heat Transfer in 3D Printing via Fused-Deposition Modeling (FDM)
by Büryan Apaçoğlu-Turan, Kadir Kırkköprü and Murat Çakan
Computation 2024, 12(2), 27; https://doi.org/10.3390/computation12020027 - 5 Feb 2024
Cited by 1 | Viewed by 2278
Abstract
Fused-Deposition Modeling (FDM) is a commonly used 3D printing method for rapid prototyping and the fabrication of plastic components. The history of temperature variation during the FDM process plays a crucial role in the degree of bonding between layers. This study presents research [...] Read more.
Fused-Deposition Modeling (FDM) is a commonly used 3D printing method for rapid prototyping and the fabrication of plastic components. The history of temperature variation during the FDM process plays a crucial role in the degree of bonding between layers. This study presents research on the thermal analysis of the 3D printing process using a developed simulation code. The code employs numerical discretization methods with an implicit scheme and an effective heat transfer coefficient for cooling. The computational model is validated by comparing the results with analytical solutions, demonstrating an agreement of more than 99%. The code is then utilized to perform thermal analyses for the 3D printing process. Interlayer and intralayer reheating effects, sensitivity to printing parameters, and realistic printing patterns are investigated. It is shown that concentric and zigzag paths yield similar peaks at different time intervals. Nodal temperatures can fall below the glass transition temperature (Tg) during the printing process, especially at the outer nodes of the domain and under conditions where the cooling period is longer and the printed volume per unit time is smaller. The article suggests future work to calculate welding time at different conditions and locations for the estimation of the degree of bonding. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Schematic representation of i,j,k indices.</p> Full article ">Figure 2
<p>Schematic representation of trajectory and dynamic boundary conditions updated with trajectory, and definition of H1, H2, H3, and H4 directions.</p> Full article ">Figure 2 Cont.
<p>Schematic representation of trajectory and dynamic boundary conditions updated with trajectory, and definition of H1, H2, H3, and H4 directions.</p> Full article ">Figure 3
<p>Temperature results obtained at the center of the benchmark geometry with different mesh sizes in x, y, and z directions.</p> Full article ">Figure 4
<p>Comparison between the computational and the analytical solutions.</p> Full article ">Figure 5
<p>Snapshots of temperatures showing change in different time intervals in 3D.</p> Full article ">Figure 6
<p>Nodal temperatures under different extrusion temperatures (at the center of 10th layer).</p> Full article ">Figure 7
<p>Nodal temperatures under different heat transfer coefficients (at the center of 10th layer).</p> Full article ">Figure 8
<p>Nodal temperatures under different air temperatures (at the center node of 10th layer).</p> Full article ">Figure 9
<p>Nodal temperature histories under different bed temperatures (at the center node of 10th layer).</p> Full article ">Figure 10
<p>Nodal temperature histories at different nodes at the same layer (4th layer).</p> Full article ">Figure 11
<p>Nodal temperature histories at different locations at the 4th layer. Dotted circle: instance of the third peak.</p> Full article ">Figure 12
<p>Onset of diffusion from 5th tour at the 5th layer to node 7 (corner node in layer 4).</p> Full article ">Figure 13
<p>Nodal temperature histories at different layers (at the center of the layer). Dotted circle: lowest temperature in a layer.</p> Full article ">Figure 14
<p>Comparison between zigzag and concentric patterns at the center node.</p> Full article ">
15 pages, 1266 KiB  
Article
Two Iterative Methods for Sizing Pipe Diameters in Gas Distribution Networks with Loops
by Dejan Brkić
Computation 2024, 12(2), 25; https://doi.org/10.3390/computation12020025 - 1 Feb 2024
Cited by 1 | Viewed by 2385
Abstract
Closed-loop pipe systems allow the possibility of the flow of gas from both directions across each route, ensuring supply continuity in the event of a failure at one point, but their main shortcoming is in the necessity to model them using iterative methods. [...] Read more.
Closed-loop pipe systems allow the possibility of the flow of gas from both directions across each route, ensuring supply continuity in the event of a failure at one point, but their main shortcoming is in the necessity to model them using iterative methods. Two iterative methods of determining the optimal pipe diameter in a gas distribution network with closed loops are described in this paper, offering the advantage of maintaining the gas velocity within specified technical limits, even during peak demand. They are based on the following: (1) a modified Hardy Cross method with the correction of the diameter in each iteration and (2) the node-loop method, which provides a new diameter directly in each iteration. The calculation of the optimal pipe diameter in such gas distribution networks relies on ensuring mass continuity at nodes, following the first Kirchhoff law, and concluding when the pressure drops in all the closed paths are algebraically balanced, adhering to the second Kirchhoff law for energy equilibrium. The presented optimisation is based on principles developed by Hardy Cross in the 1930s for the moment distribution analysis of statically indeterminate structures. The results are for steady-state conditions and for the highest possible estimated demand of gas, while the distributed gas is treated as a noncompressible fluid due to the relatively small drop in pressure in a typical network of pipes. There is no unique solution; instead, an infinite number of potential outcomes exist, alongside infinite combinations of pipe diameters for a given fixed flow pattern that can satisfy the first and second Kirchhoff laws in the given topology of the particular network at hand. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Differences between approaches of the two proposed loop-oriented methods for optimisation: (<b>a</b>) diameter correction D = D<sub>i−1</sub> + ΔD, Hardy Cross method—Brkić (2009) [<a href="#B5-computation-12-00025" class="html-bibr">5</a>] and Corfield et al. [<a href="#B7-computation-12-00025" class="html-bibr">7</a>]; (b) direct calculation of D, node-loop method.</p> Full article ">Figure 2
<p>Illustrative network of pipes with loops (black arrows represent inputs and outputs of the network, while white arrows are flows through pipes).</p> Full article ">Figure 3
<p>Possible extrema of the optimisation function—general illustrative example.</p> Full article ">
17 pages, 381 KiB  
Article
Exploring Controlled Passive Particle Motion Driven by Point Vortices on a Sphere
by Carlos Balsa, M. Victoria Otero-Espinar and Sílvio Gama
Computation 2024, 12(2), 23; https://doi.org/10.3390/computation12020023 - 31 Jan 2024
Cited by 1 | Viewed by 1600
Abstract
This work focuses on optimizing the displacement of a passive particle interacting with vortices located on the surface of a sphere. The goal is to minimize the energy expended during the displacement within a fixed time. The modeling of particle dynamics, whether in [...] Read more.
This work focuses on optimizing the displacement of a passive particle interacting with vortices located on the surface of a sphere. The goal is to minimize the energy expended during the displacement within a fixed time. The modeling of particle dynamics, whether in Cartesian or spherical coordinates, gives rise to alternative formulations of the identical problem. Thanks to these two versions of the same problem, we can assert that the algorithm, employed to transform the optimal control problem into an optimization problem, is effective, as evidenced by the obtained controls. The numerical resolution of these formulations through a direct approach consistently produces optimal solutions, regardless of the selected coordinate system. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Trajectories resulting from the solutions of the optimization problems DP1, DP2, and DP3 with a set of <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> controls.</p> Full article ">Figure 2
<p>Trajectories corresponding to the minimum values of the objective function in the case of a flow induced by two vortices.</p> Full article ">Figure 3
<p>Trajectories corresponding to the minimum values of the objective function in the case of a flow induced by three vortices.</p> Full article ">
23 pages, 441 KiB  
Article
Maxwell’s True Current
by Robert S. Eisenberg
Computation 2024, 12(2), 22; https://doi.org/10.3390/computation12020022 - 31 Jan 2024
Viewed by 1758
Abstract
Maxwell defined a ‘true’ or ‘total’ current in a way not widely used today. He said that “… true electric current … is not the same thing as the current of conduction but that the time-variation of the electric displacement must be taken [...] Read more.
Maxwell defined a ‘true’ or ‘total’ current in a way not widely used today. He said that “… true electric current … is not the same thing as the current of conduction but that the time-variation of the electric displacement must be taken into account in estimating the total movement of electricity”. We show that the true or total current is a universal property of electrodynamics independent of the properties of matter. We use mathematics without the approximation of a dielectric constant. The resulting Maxwell current law is a generalization of the Kirchhoff law of current used in circuit analysis, that also includes the displacement current. The generalization is not a long-time low-frequency approximation in contrast to the traditional presentation of Kirchhoff’s law. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
22 pages, 3189 KiB  
Article
A Technical Comparative Heart Disease Prediction Framework Using Boosting Ensemble Techniques
by Najmu Nissa, Sanjay Jamwal and Mehdi Neshat
Computation 2024, 12(1), 15; https://doi.org/10.3390/computation12010015 - 16 Jan 2024
Cited by 2 | Viewed by 3209
Abstract
This paper addresses the global surge in heart disease prevalence and its impact on public health, stressing the need for accurate predictive models. The timely identification of individuals at risk of developing cardiovascular ailments is paramount for implementing preventive measures and timely interventions. [...] Read more.
This paper addresses the global surge in heart disease prevalence and its impact on public health, stressing the need for accurate predictive models. The timely identification of individuals at risk of developing cardiovascular ailments is paramount for implementing preventive measures and timely interventions. The World Health Organization (WHO) reports that cardiovascular diseases, responsible for an alarming 17.9 million annual fatalities, constitute a significant 31% of the global mortality rate. The intricate clinical landscape, characterized by inherent variability and a complex interplay of factors, poses challenges for accurately diagnosing the severity of cardiac conditions and predicting their progression. Consequently, early identification emerges as a pivotal factor in the successful treatment of heart-related ailments. This research presents a comprehensive framework for the prediction of cardiovascular diseases, leveraging advanced boosting techniques and machine learning methodologies, including Cat boost, Random Forest, Gradient boosting, Light GBM, and Ada boost. Focusing on “Early Heart Disease Prediction using Boosting Techniques”, this paper aims to contribute to the development of robust models capable of reliably forecasting cardiovascular health risks. Model performance is rigorously assessed using a substantial dataset on heart illnesses from the UCI machine learning library. With 26 feature-based numerical and categorical variables, this dataset encompasses 8763 samples collected globally. The empirical findings highlight AdaBoost as the preeminent performer, achieving a notable accuracy of 95% and excelling in metrics such as negative predicted value (0.83), false positive rate (0.04), false negative rate (0.04), and false development rate (0.01). These results underscore AdaBoost’s superiority in predictive accuracy and overall performance compared to alternative algorithms, contributing valuable insights to the field of cardiovascular health prediction. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Frequency distribution of classes in the heart disease dataset.</p> Full article ">Figure 2
<p>The distribution of the heart dataset features.</p> Full article ">Figure 3
<p>The relationships and patterns within the heart disease dataset.</p> Full article ">Figure 4
<p>The details of methodology adapted.</p> Full article ">Figure 5
<p>The details of pre-processing steps.</p> Full article ">Figure 6
<p>The components of the confusion matrix.</p> Full article ">Figure 7
<p>Ten-fold cross validation technique.</p> Full article ">Figure 8
<p>The testing confusion matrix of (<b>a</b>) GBoost, (<b>b</b>) AdaBoost, (<b>c</b>) CatBoost, (<b>d</b>) Light Boost, (<b>e</b>) Random Forest, (<b>f</b>) XGBoost, and (<b>g</b>) Decision Tree.</p> Full article ">Figure 9
<p>Features importance based on the AdaBoost.</p> Full article ">
21 pages, 5167 KiB  
Article
Enhancement of Machine-Learning-Based Flash Calculations near Criticality Using a Resampling Approach
by Eirini Maria Kanakaki, Anna Samnioti and Vassilis Gaganis
Computation 2024, 12(1), 10; https://doi.org/10.3390/computation12010010 - 9 Jan 2024
Cited by 4 | Viewed by 2098
Abstract
Flash calculations are essential in reservoir engineering applications, most notably in compositional flow simulation and separation processes, to provide phase distribution factors, known as k-values, at a given pressure and temperature. The calculation output is subsequently used to estimate composition-dependent properties of interest, [...] Read more.
Flash calculations are essential in reservoir engineering applications, most notably in compositional flow simulation and separation processes, to provide phase distribution factors, known as k-values, at a given pressure and temperature. The calculation output is subsequently used to estimate composition-dependent properties of interest, such as the equilibrium phases’ molar fraction, composition, density, and compressibility. However, when the flash conditions approach criticality, minor inaccuracies in the computed k-values may lead to significant deviation in the dependent properties, which is eventually inherited to the simulator, leading to large errors in the simulation. Although several machine-learning-based regression approaches have emerged to drastically accelerate flash calculations, the criticality issue persists. To address this problem, a novel resampling technique of the ML models’ training data population is proposed, which aims to fine-tune the training dataset distribution and optimally exploit the models’ learning capacity across various flash conditions. The results demonstrate significantly improved accuracy in predicting phase behavior results near criticality, offering valuable contributions not only to the subsurface reservoir engineering industry but also to the broader field of thermodynamics. By understanding and optimizing the model’s training, this research enables more precise predictions and better-informed decision-making processes in domains involving phase separation phenomena. The proposed technique is applicable to every ML-dominated regression problem, where properties dependent on the machine output are of interest rather than the model output itself. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Reservoir model consisting of millions of grid blocks [<a href="#B6-computation-12-00010" class="html-bibr">6</a>].</p> Full article ">Figure 2
<p>Schematic representation of the flash problem.</p> Full article ">Figure 3
<p>Effect of pressure on k-values at reservoir temperature in the 1500 to 4054.3 psi range.</p> Full article ">Figure 4
<p>Regular phase envelope and shadow region of a gas condensate.</p> Full article ">Figure 5
<p>Phase envelope and convergence locus of the lean (on the <b>left</b>) and rich (on the <b>right</b>) gas condensates.</p> Full article ">Figure 6
<p>Pressure and k-value histograms of the base training dataset.</p> Full article ">Figure 7
<p>Average absolute errors in conventional ML model training for each dependent property and datapoint class.</p> Full article ">Figure 8
<p>Outline of the proposed resampling algorithm.</p> Full article ">Figure 9
<p>Number of datapoints in the balanced dataset across the 10 classes with respect to hyperparameter <math display="inline"><semantics> <mrow> <mi>D</mi> </mrow> </semantics></math>, based on <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mi>L</mi> </mrow> </msub> </mrow> </semantics></math> error resampling.</p> Full article ">Figure 10
<p>Improvement/decline in the average absolute error and standard deviation versus <math display="inline"><semantics> <mrow> <mi>D</mi> </mrow> </semantics></math> based on <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mi>L</mi> </mrow> </msub> </mrow> </semantics></math> error resampling.</p> Full article ">Figure 11
<p>Κ-value histograms of the balanced training dataset.</p> Full article ">Figure 12
<p>Comparison of absolute average error in <math display="inline"><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math> per class using the base training dataset and the resampled training dataset.</p> Full article ">Figure 13
<p>Comparison of absolute average error in <math display="inline"><semantics> <mrow> <mi mathvariant="bold">x</mi> </mrow> </semantics></math> per class using the base training dataset and the resampled training dataset.</p> Full article ">Figure 14
<p>Comparison of absolute average error in <math display="inline"><semantics> <mrow> <mi mathvariant="bold">y</mi> </mrow> </semantics></math> per class using the base training dataset and the resampled training dataset.</p> Full article ">Figure 15
<p>Comparison of absolute average error in <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mi>L</mi> </mrow> </msub> </mrow> </semantics></math> per class using the base training dataset and the resampled training dataset.</p> Full article ">Figure 16
<p>Comparison of absolute average error in <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mi>V</mi> </mrow> </msub> </mrow> </semantics></math> per class using the base training dataset and the resampled training dataset.</p> Full article ">Figure 17
<p>Improvement/decline in the average absolute error and standard deviation versus <math display="inline"><semantics> <mrow> <mi>D</mi> </mrow> </semantics></math> based on <math display="inline"><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math> error resampling.</p> Full article ">
16 pages, 4913 KiB  
Article
LSTM Reconstruction of Turbulent Pressure Fluctuation Signals
by Konstantinos Poulinakis, Dimitris Drikakis, Ioannis W. Kokkinakis, S. Michael Spottswood and Talib Dbouk
Computation 2024, 12(1), 4; https://doi.org/10.3390/computation12010004 - 1 Jan 2024
Viewed by 2239
Abstract
This paper concerns the application of a long short-term memory model (LSTM) for high-resolution reconstruction of turbulent pressure fluctuation signals from sparse (reduced) data. The model’s training was performed using data from high-resolution computational fluid dynamics (CFD) simulations of high-speed turbulent boundary layers [...] Read more.
This paper concerns the application of a long short-term memory model (LSTM) for high-resolution reconstruction of turbulent pressure fluctuation signals from sparse (reduced) data. The model’s training was performed using data from high-resolution computational fluid dynamics (CFD) simulations of high-speed turbulent boundary layers over a flat panel. During the preprocessing stage, we employed cubic spline functions to increase the fidelity of the sparse signals and subsequently fed them to the LSTM model for a precise reconstruction. We evaluated our reconstruction method with the root mean squared error (RMSE) metric and via inspection of power spectrum plots. Our study reveals that the model achieved a precise high-resolution reconstruction of the training signal and could be transferred to new unseen signals of a similar nature with extremely high success. The numerical simulations show promising results for complex turbulent signals, which may be experimentally or computationally produced. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Vorticity contour plot near the wall in the fully turbulent region. The <span class="html-italic">x</span> axis indicates the distance from the plate’s leading edge, and the <span class="html-italic">y</span> axis is the spanwise position, with both shown in reduced units (normalized by <math display="inline"><semantics> <msub> <mi>x</mi> <mi>l</mi> </msub> </semantics></math>). The numbered black stars indicate the probes’ positions to calculate the pressure fluctuations induced by the flow on the wall.</p> Full article ">Figure 2
<p>Illustration of pressure fluctuation for probe 2 in the time domain: (<b>top</b>) original signal, (<b>middle</b>) sparsity = 40, and (<b>bottom</b>) sparsity = 100. All values are dimensionless.</p> Full article ">Figure 3
<p>High-level schematic of the neural network architecture. The length of the sequence L is dynamic and depends on the sampling step <math display="inline"><semantics> <mrow> <mo>ℓ</mo> <mo>=</mo> <mi>m</mi> <mo>∗</mo> <mi>s</mi> <mi>p</mi> <mi>a</mi> <mi>r</mi> <mi>s</mi> <mi>i</mi> <mi>t</mi> <mi>y</mi> <mo>_</mo> <mi>f</mi> <mi>a</mi> <mi>c</mi> <mi>t</mi> <mi>o</mi> <mi>r</mi> </mrow> </semantics></math>. The symbols <math display="inline"><semantics> <msub> <mi>h</mi> <mi>t</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>c</mi> <mi>t</mi> </msub> </semantics></math> denote the LSTM cell’s hidden state and cell state outputs, respectively. The model’s inputs and output are <math display="inline"><semantics> <msub> <mi>X</mi> <mi>t</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>X</mi> <mrow> <mi>L</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </semantics></math>, respectively.</p> Full article ">Figure 4
<p>Predictions of the hybrid LSTM model at probe 1 for a sparsity factor equal to 5. We zoomed in on random samples to better visualize and evaluate the reconstruction error.</p> Full article ">Figure 5
<p>Predictions of the hybrid LSTM model at probe 1 for a sparsity factor equal to 40. We zoomed in on random samples to better visualize and evaluate the reconstruction error.</p> Full article ">Figure 6
<p>Predictions of the hybrid LSTM model at probe 1 for a sparsity factor equal to 100. We zoomed in on random samples to better visualize and evaluate the reconstruction error.</p> Full article ">Figure 7
<p>Predictions of the hybrid LSTM model at probe 4 for a sparsity factor equal to 100. We zoomed in on random samples to better visualize and evaluate the reconstruction error.</p> Full article ">Figure 8
<p>Predictions of the hybrid LSTM model at probe 9 for a sparsity factor equal to 100. We zoomed in on random samples to better visualize and evaluate the reconstruction error.</p> Full article ">Figure 9
<p>Predictions of the hybrid LSTM model at probe 10 for a sparsity factor equal to 100. We zoomed in on random samples to better visualize and evaluate the reconstruction error.</p> Full article ">Figure 10
<p>Normalized RMSE % vs. the sparsity factor used to sample the training dataset.</p> Full article ">Figure 11
<p>Power spectrum of the original signal (ground truth) and the LSTM prediction for different sparsities at probes (from top to bottom and left to right) 1, 2 (training point), 4, 9, and 10. The <span class="html-italic">x</span> axis is the Strouhal number, and the <span class="html-italic">y</span> axis is the power spectrum in terms of pascals squared.</p> Full article ">
20 pages, 2411 KiB  
Article
MSVR & Operator-Based System Design of Intelligent MIMO Sensorless Control for Microreactor Devices
by Tatsuma Kato, Kosuke Nishizawa and Mingcong Deng
Computation 2024, 12(1), 2; https://doi.org/10.3390/computation12010002 - 25 Dec 2023
Viewed by 1919
Abstract
Recently, microreactors, which are tubular reactors capable of fast and highly efficient chemical reactions, have attracted attention. However, precise temperature control is required because temperature changes due to reaction heat can cause reactions to proceed differently from those designed. In a previous study, [...] Read more.
Recently, microreactors, which are tubular reactors capable of fast and highly efficient chemical reactions, have attracted attention. However, precise temperature control is required because temperature changes due to reaction heat can cause reactions to proceed differently from those designed. In a previous study, a single-input/output nonlinear control system was proposed using a model in which the microreactor is divided into three regions and the thermal equation is formulated considering the temperature gradient, but it could not control two different temperatures. In this paper, a multi-input, multi-output nonlinear control system was designed using operator theory. On the other hand, when the number of parallel microreactors is increased, a sensorless control method using M–SVR with a generalized Gaussian kernel was incorporated into the MIMO nonlinear control system from the viewpoint of cost reduction, and the effectiveness of the proposed method was confirmed via experimental results. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Model of microreactor and heat spreader.</p> Full article ">Figure 2
<p>Without interference effects.</p> Full article ">Figure 3
<p>MIMO control system for microreactor based on operator theory.</p> Full article ">Figure 4
<p>Proposed sensorless control system using M–SVR.</p> Full article ">Figure 5
<p>Diagram of the experimental apparatus.</p> Full article ">Figure 6
<p>Microreactor temperature (simulation on MIMO control system).</p> Full article ">Figure 7
<p>Control input (simulation on MIMO control system).</p> Full article ">Figure 8
<p>Microreactor temperature (experiment on MIMO control system).</p> Full article ">Figure 9
<p>Control input (experiment on MIMO control system).</p> Full article ">Figure 10
<p>Microreactor temperature <math display="inline"><semantics> <msub> <mi>T</mi> <msub> <mi>w</mi> <mn>1</mn> </msub> </msub> </semantics></math> and M–SVR estimate <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mi>M</mi> <msub> <mi>w</mi> <mn>1</mn> </msub> </mrow> </msub> </semantics></math> (experiment).</p> Full article ">Figure 11
<p>Microreactor temperature <math display="inline"><semantics> <msub> <mi>T</mi> <msub> <mi>w</mi> <mn>3</mn> </msub> </msub> </semantics></math> and M–SVR estimate <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mi>M</mi> <msub> <mi>w</mi> <mn>3</mn> </msub> </mrow> </msub> </semantics></math> (experiment).</p> Full article ">Figure 12
<p>Control input (experiment on MIMO sensorless control using M–SVR).</p> Full article ">Figure A1
<p>MIMO system plant.</p> Full article ">Figure A2
<p>MIMO system stabilization via operator theory.</p> Full article ">
34 pages, 5630 KiB  
Article
Shear-Enhanced Compaction Analysis of the Vaca Muerta Formation
by José G. Hasbani, Evan M. C. Kias, Roberto Suarez-Rivera and Victor M. Calo
Computation 2023, 11(12), 250; https://doi.org/10.3390/computation11120250 - 10 Dec 2023
Cited by 1 | Viewed by 2156
Abstract
The laboratory measurements conducted on Vaca Muerta formation samples demonstrate stress-dependent elastic behavior and compaction under representative in situ conditions. The experimental results reveal that the analyzed samples display elastoplastic deformation and shear-enhanced compaction as primary plasticity mechanisms. These experimental findings contradict the [...] Read more.
The laboratory measurements conducted on Vaca Muerta formation samples demonstrate stress-dependent elastic behavior and compaction under representative in situ conditions. The experimental results reveal that the analyzed samples display elastoplastic deformation and shear-enhanced compaction as primary plasticity mechanisms. These experimental findings contradict the expected linear elastic response anticipated before brittle failure, as reported in several studies on the geomechanical characterization of the Vaca Muerta formation. Therefore, we present a comprehensive laboratory analysis of Vaca Muerta formation samples showing their nonlinear elastic behavior and irrecoverable shear-enhanced compaction. Additionally, we calibrate an elastoplastic constitutive model based on these experimental observations. The resulting model accurately reproduces the observed phenomena, playing a pivotal role in geoengineering applications within the energy industry. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Triaxial GCST RTR-1000 testing system (<a href="https://www.gcts.com/" target="_blank">https://www.gcts.com/</a>).</p> Full article ">Figure 2
<p>General laboratory test program for Vaca Muerta samples.</p> Full article ">Figure 3
<p>Additive decomposition of a porous medium volume.</p> Full article ">Figure 4
<p>Evolution of volumetric strain as a function of mean stress during drained hydrostatic test on Sample 0.</p> Full article ">Figure 5
<p>Evolution of porosity as a function of mean stress during drained hydrostatic test on Sample 0.</p> Full article ">Figure 6
<p>Elastic constant determination for Sample 1. (<b>Upper left</b>) Young’s modulus determination. (<b>Upper right</b>) Bulk modulus determination. (<b>Bottom left</b>) Poisson’s ratio determination. (<b>Bottom right</b>) Volumetric strain evolution during drained triaxial test.</p> Full article ">Figure 7
<p>Elastic constant determination for Sample 2. (<b>Upper left</b>) Young’s modulus determination. (<b>Upper right</b>) Bulk modulus determination. (<b>Bottom left</b>) Poisson’s ratio determination. (<b>Bottom right</b>) Volumetric strain evolution during drained triaxial test.</p> Full article ">Figure 8
<p>Elastic constant determination for Sample 3. (<b>Upper left</b>) Young’s modulus determination. (<b>Upper right</b>) Bulk modulus determination. (<b>Bottom left</b>) Poisson’s ratio determination. (<b>Bottom right</b>) Volumetric strain evolution during drained triaxial test.</p> Full article ">Figure 9
<p>Elastic constant determination for Sample 4. (<b>Upper left</b>) Young’s modulus determination. (<b>Upper right</b>) Bulk modulus determination. (<b>Bottom left</b>) Poisson’s ratio determination. (<b>Bottom right</b>) Volumetric strain evolution during drained triaxial test.</p> Full article ">Figure 10
<p>Shear-enhanced compaction analysis for Sample 1. (<b>Left</b>) Volumetric strain evolution as a function of mean pressure. (<b>Right</b>) Porosity evolution as a function of mean stress.</p> Full article ">Figure 11
<p>Shear-enhanced compaction analysis for Sample 2. (<b>Left</b>) Volumetric strain evolution as a function of mean pressure. (<b>Right</b>) Porosity evolution as a function of mean stress.</p> Full article ">Figure 12
<p>Shear-enhanced compaction analysis for Sample 3. (<b>Left</b>) Volumetric strain evolution as a function of mean pressure. (<b>Right</b>) Porosity evolution as a function of mean stress.</p> Full article ">Figure 13
<p>Shear-enhanced compaction analysis for Sample 4. (<b>Left</b>) Volumetric strain evolution as a function of mean pressure. (<b>Right</b>) Porosity evolution as a function of mean stress.</p> Full article ">Figure 14
<p>Representation of the Cam-Clay yield criterion in the <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>−</mo> <mi>q</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 15
<p>Compaction and bulk volumetric recovery parameters from hydrostatic cycling.</p> Full article ">Figure 16
<p>Porosity degradation parameter determination.</p> Full article ">Figure 17
<p>Determination of the critical state line slope <span class="html-italic">M</span> from drained triaxial tests.</p> Full article ">Figure 18
<p>Internal variables updated by the closest point projection algorithm. The blue point is the trial state for the internal variables (elastic prediction). The red dot is the final state for the internal variables (plastic correction).</p> Full article ">Figure 19
<p>Comparison between numerical simulation of a triaxial test and laboratory tests. (<b>Left</b>) Deviatoric stress against axial strain. (<b>Right</b>) Mean stress against volumetric strain.</p> Full article ">
18 pages, 6424 KiB  
Article
Effects of Running in Minimal, Maximal and Conventional Footwear on Tibial Stress Fracture Probability: An Examination Using Finite Element and Probabilistic Analyses
by Jonathan Sinclair and Paul John Taylor
Computation 2023, 11(12), 248; https://doi.org/10.3390/computation11120248 - 6 Dec 2023
Cited by 1 | Viewed by 2879
Abstract
This study examined the effects of minimal, maximal and conventional running footwear on tibial strains and stress fracture probability using finite element and probabilistic analyses. The current investigation examined fifteen males running in three footwear conditions (minimal, maximal and conventional). Kinematic data were [...] Read more.
This study examined the effects of minimal, maximal and conventional running footwear on tibial strains and stress fracture probability using finite element and probabilistic analyses. The current investigation examined fifteen males running in three footwear conditions (minimal, maximal and conventional). Kinematic data were collected during overground running at 4.0 m/s using an eight-camera motion-capture system and ground reaction forces using a force plate. Tibial strains were quantified using finite element modelling and stress fracture probability calculated via probabilistic modelling over 100 days of running. Ninetieth percentile tibial strains were significantly greater in minimal (4681.13 με) (p < 0.001) and conventional (4498.84 με) (p = 0.007) footwear compared to maximal (4069.65 με). Furthermore, tibial stress fracture probability was significantly greater in minimal footwear (0.22) (p = 0.047) compared to maximal (0.15). The observations from this investigation show that compared to minimal footwear, maximal running shoes appear to be effective in attenuating runners’ likelihood of developing a tibial stress fracture. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Experimental footwear (<b>a</b>) conventional, (<b>b</b>) maximal and (<b>c</b>) minimal.</p> Full article ">Figure 2
<p>(<b>a</b>) Experimental retroreflective marker positions and (<b>b</b>) segment co-ordinate systems (R = right and L = left, TR = trunk, P = pelvis, T = thigh, S = shank and F = foot, X = sagittal plane, Y = coronal plane and Z = transverse plane).</p> Full article ">Figure 3
<p>OpenSim musculoskeletal simulation model.</p> Full article ">Figure 4
<p>Depiction of finite element model mesh with loading and boundary conditions. The tibial plateau was fully constrained (<b>a</b>). Ankle joint contact forces were applied to the distal tibia (<b>b</b>); muscle forces (not all shown here) were applied as concentrated forces at their insertion point onto the tibia (<b>c</b>), and residual moments were applied at the distal tibia (<b>d</b>).</p> Full article ">Figure 5
<p>Representative tibial strain distribution on the tibia.</p> Full article ">Figure 6
<p>Average probabilities of failure (PF<sub>RA</sub>) in each footwear condition across 100 days of running.</p> Full article ">
26 pages, 7420 KiB  
Article
Design and Implementation of a Camera-Based Tracking System for MAV Using Deep Learning Algorithms
by Stefan Hensel, Marin B. Marinov and Raphael Panter
Computation 2023, 11(12), 244; https://doi.org/10.3390/computation11120244 - 4 Dec 2023
Viewed by 2308
Abstract
In recent years, the advancement of micro-aerial vehicles has been rapid, leading to their widespread utilization across various domains due to their adaptability and efficiency. This research paper focuses on the development of a camera-based tracking system specifically designed for low-cost drones. The [...] Read more.
In recent years, the advancement of micro-aerial vehicles has been rapid, leading to their widespread utilization across various domains due to their adaptability and efficiency. This research paper focuses on the development of a camera-based tracking system specifically designed for low-cost drones. The primary objective of this study is to build up a system capable of detecting objects and locating them on a map in real time. Detection and positioning are achieved solely through the utilization of the drone’s camera and sensors. To accomplish this goal, several deep learning algorithms are assessed and adopted because of their suitability with the system. Object detection is based upon a single-shot detector architecture chosen for maximum computation speed, and the tracking is based upon the combination of deep neural-network-based features combined with an efficient sorting strategy. Subsequently, the developed system is evaluated using diverse metrics to determine its performance for detection and tracking. To further validate the approach, the system is employed in the real world to show its possible deployment. For this, two distinct scenarios were chosen to adjust the algorithms and system setup: a search and rescue scenario with user interaction and precise geolocalization of missing objects, and a livestock control scenario, showing the capability of surveying individual members and keeping track of number and area. The results demonstrate that the system is capable of operating in real time, and the evaluation verifies that the implemented system enables precise and reliable determination of detected object positions. The ablation studies prove that object identification through small variations in phenotypes is feasible with our approach. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Communication within the system.</p> Full article ">Figure 2
<p>Timing of communication between client and server during a system scan.</p> Full article ">Figure 3
<p>User interface mockup.</p> Full article ">Figure 4
<p>User interface implementation.</p> Full article ">Figure 5
<p>Overview of detected objects on the map. Red symbol indicates the starting position, blue ones are detected objects, the orange symbol is a manually selected object.</p> Full article ">Figure 6
<p>Definition of the flight area. Starting position of drone in red, blue markers indicate manually selected polygonal points.</p> Full article ">Figure 7
<p>Building the grids within the bounding box. Yellow polygon in cartesian coordinates, the blue grid and center points are in ellipsoidal coordinates for GNSS (WGS84).</p> Full article ">Figure 8
<p>Determination of the flight path points within the bounding box (<b>a</b>) Definition of the individual points; (<b>b</b>) Defined points within the flight range.</p> Full article ">Figure 9
<p>Final flight path in defined polygon, marked in red.</p> Full article ">Figure 10
<p>A defined area for simulating a search-and-rescue scenario.</p> Full article ">Figure 11
<p>Confusion matrix of the trained model.</p> Full article ">Figure 12
<p><math display="inline"><semantics> <mrow> <msub> <mrow> <mi>F</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math>-score curve of the trained YOLOv5 model.</p> Full article ">Figure 13
<p>Overview of different metrics for the trained YOLOv5 model.</p> Full article ">Figure 14
<p>Comparison between an original teddy bear and an edited teddy bear: (<b>a</b>) Original teddy bear; (<b>b</b>) Modified teddy bear with randomly generated points. Coverage: ~10.2%.</p> Full article ">Figure 15
<p>Results of the Cosine Similarity calculations.</p> Full article ">Figure 16
<p>Influence of image brightness on Cosine Similarity.</p> Full article ">Figure 17
<p>Combination of brightness, generated points, and background change.</p> Full article ">Figure 18
<p>Best Precision/Recall values of all trained detectors.</p> Full article ">
24 pages, 571 KiB  
Article
Wind Farm Cable Connection Layout Optimization Using a Genetic Algorithm and Integer Linear Programming
by Eduardo J. Solteiro Pires, Adelaide Cerveira and José Baptista
Computation 2023, 11(12), 241; https://doi.org/10.3390/computation11120241 - 3 Dec 2023
Cited by 2 | Viewed by 2081
Abstract
This work addresses the wind farm (WF) optimization layout considering several substations. It is given a set of wind turbines jointly with a set of substations, and the goal is to obtain the optimal design to minimize the infrastructure cost and the cost [...] Read more.
This work addresses the wind farm (WF) optimization layout considering several substations. It is given a set of wind turbines jointly with a set of substations, and the goal is to obtain the optimal design to minimize the infrastructure cost and the cost of electrical energy losses during the wind farm lifetime. The turbine set is partitioned into subsets to assign to each substation. The cable type and the connections to collect wind turbine-produced energy, forwarding to the corresponding substation, are selected in each subset. The technique proposed uses a genetic algorithm (GA) and an integer linear programming (ILP) model simultaneously. The GA creates a partition in the turbine set and assigns each of the obtained subsets to a substation to optimize a fitness function that corresponds to the minimum total cost of the WF layout. The fitness function evaluation requires solving an ILP model for each substation to determine the optimal cable connection layout. This methodology is applied to four onshore WFs. The obtained results show that the solution performance of the proposed approach reaches up to 0.17% of economic savings when compared to the clustering with ILP approach (an exact approach). Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Wind farm layouts: (<b>a</b>) with several clusters and substations, (<b>b</b>) with only one substation.</p> Full article ">Figure 2
<p>Line diagram of a radial distribution system.</p> Full article ">Figure 3
<p>Example of a wind farm layout with two main branches (blue WT {1,3,4} and red WT {2,5,6,7,8}). Some branches show the current flowing through them.</p> Full article ">Figure 4
<p>Chromosome representation.</p> Full article ">Figure 5
<p>Crossover operator.</p> Full article ">Figure 6
<p>Turbines and cable connection layout for the <span class="html-italic">Alto da Coutada</span> wind farm. In blue are the WTs connected to Substation <math display="inline"><semantics> <msub> <mi>O</mi> <mn>1</mn> </msub> </semantics></math> and in orange are the WTs connected to Substation <math display="inline"><semantics> <msub> <mi>O</mi> <mn>2</mn> </msub> </semantics></math>. In blue with orange borders are the WTs that would be connected to Substation <math display="inline"><semantics> <msub> <mi>O</mi> <mn>2</mn> </msub> </semantics></math> using the clustering method.</p> Full article ">Figure 7
<p>Turbines and the cable connection layout for the WF-3S wind farm. The WTs connected to substations <math display="inline"><semantics> <msub> <mi>O</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>O</mi> <mn>2</mn> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>O</mi> <mn>3</mn> </msub> </semantics></math> are shown in green, red, and blue, respectively. In blue with a green border are the WTs that would be connected to Substation <math display="inline"><semantics> <msub> <mi>O</mi> <mn>1</mn> </msub> </semantics></math> using the clustering method.</p> Full article ">Figure 8
<p>Turbines and cable connection layout for the WF-S4 wind farm. The WTs connected to substations <math display="inline"><semantics> <msub> <mi>O</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>O</mi> <mn>2</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>O</mi> <mn>3</mn> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>O</mi> <mn>4</mn> </msub> </semantics></math> are shown in orange, red, green, and blue, respectively. Using the clustering method, WTs 8 and 40 should be connected to <math display="inline"><semantics> <msub> <mi>O</mi> <mn>1</mn> </msub> </semantics></math>, and WT 73 should be connected to <math display="inline"><semantics> <msub> <mi>O</mi> <mn>4</mn> </msub> </semantics></math>.</p> Full article ">Figure 9
<p>Turbines and cable connection layout for the <span class="html-italic">Alto Minho</span> wind farm of the (<b>a</b>) entire park, (<b>b</b>) <span class="html-italic">Picos</span> wind field, (<b>c</b>) <span class="html-italic">São Silvestre</span> wind field, (<b>d</b>) <span class="html-italic">Mendoiro</span> wind field, (<b>e</b>) <span class="html-italic">Santo António</span> I wind field, and (<b>f</b>) <span class="html-italic">Santo António</span> II wind field.</p> Full article ">Figure 9 Cont.
<p>Turbines and cable connection layout for the <span class="html-italic">Alto Minho</span> wind farm of the (<b>a</b>) entire park, (<b>b</b>) <span class="html-italic">Picos</span> wind field, (<b>c</b>) <span class="html-italic">São Silvestre</span> wind field, (<b>d</b>) <span class="html-italic">Mendoiro</span> wind field, (<b>e</b>) <span class="html-italic">Santo António</span> I wind field, and (<b>f</b>) <span class="html-italic">Santo António</span> II wind field.</p> Full article ">
14 pages, 18797 KiB  
Article
Effects of the Number of Classes and Pressure Map Resolution on Fine-Grained In-Bed Posture Classification
by Luís Fonseca, Fernando Ribeiro and José Metrôlho
Computation 2023, 11(12), 239; https://doi.org/10.3390/computation11120239 - 2 Dec 2023
Cited by 1 | Viewed by 1783
Abstract
In-bed posture classification has attracted considerable research interest and has significant potential to enhance healthcare applications. Recent works generally use approaches based on pressure maps, machine learning algorithms and focused mainly on finding solutions to obtain high accuracy in posture classification. Typically, these [...] Read more.
In-bed posture classification has attracted considerable research interest and has significant potential to enhance healthcare applications. Recent works generally use approaches based on pressure maps, machine learning algorithms and focused mainly on finding solutions to obtain high accuracy in posture classification. Typically, these solutions use different datasets with varying numbers of sensors and classify the four main postures (supine, prone, left-facing, and right-facing) or, in some cases, include some variants of those main postures. Following this, this article has three main objectives: fine-grained detection of postures of bedridden people, identifying a large number of postures, including small variations—consideration of 28 different postures will help to better identify the actual position of the bedridden person with a higher accuracy. The number of different postures in this approach is considerably higher than the of those used in any other related work; analyze the impact of pressure map resolution on the posture classification accuracy, which has also not been addressed in other studies; and use the PoPu dataset, a dataset that includes pressure maps from 60 participants and 28 different postures. The dataset was analyzed using five distinct ML algorithms (k-nearest neighbors, linear support vector machines, decision tree, random forest, and multi-layer perceptron). This study’s findings show that the used algorithms achieve high accuracy in 4-posture classification (up to 99% in the case of MLP) using the PoPu dataset, with lower accuracies when attempting the finer-grained 28-posture classification approach (up to 68% in the case of random forest). The results indicate that using ML algorithms for finer-grained applications is possible to specify the patient’s exact position to some degree since the parent posture is still accurately classified. Furthermore, reducing the resolution of the pressure maps seems to affect the classifiers only slightly, which suggests that for applications that do not need finer-granularity, a lower resolution might suffice. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Confusion matrix of MLP classification with 4 classes.</p> Full article ">Figure 2
<p>Confusion matrix of MLP classification with 28 classes.</p> Full article ">Figure 3
<p>Tenfold cross validation accuracy results.</p> Full article ">Figure 4
<p>Resulting images from pressure values with different resolutions. From left to right: First 64 × 27 matrix; Second 32 × 14 matrix; Third 16 × 7 matrix; Fourth 8 × 4 matrix.</p> Full article ">Figure 5
<p>Confusion matrix of MLP classification with 4 classes trained using reduced 8 × 4 matrix.</p> Full article ">
15 pages, 585 KiB  
Article
Development of AI-Based Tools for Power Generation Prediction
by Ana Paula Aravena-Cifuentes, Jose David Nuñez-Gonzalez, Andoni Elola and Malinka Ivanova
Computation 2023, 11(11), 232; https://doi.org/10.3390/computation11110232 - 16 Nov 2023
Cited by 1 | Viewed by 2152
Abstract
This study presents a model for predicting photovoltaic power generation based on meteorological, temporal and geographical variables, without using irradiance values, which have traditionally posed challenges and difficulties for accurate predictions. Validation methods and evaluation metrics are used to analyse four different approaches [...] Read more.
This study presents a model for predicting photovoltaic power generation based on meteorological, temporal and geographical variables, without using irradiance values, which have traditionally posed challenges and difficulties for accurate predictions. Validation methods and evaluation metrics are used to analyse four different approaches that vary in the distribution of the training and test database, and whether or not location-independent modelling is performed. The coefficient of determination, R2, is used to measure the proportion of variation in photovoltaic power generation that can be explained by the model’s variables, while gCO2eq represents the amount of CO2 emissions equivalent to each unit of power generation. Both are used to compare model performance and environmental impact. The results show significant differences between the locations, with substantial improvements in some cases, while in others improvements are limited. The importance of customising the predictive model for each specific location is emphasised. Furthermore, it is concluded that environmental impact studies in model production are an additional step towards the creation of more sustainable and efficient models. Likewise, this research considers both the accuracy of solar energy predictions and the environmental impact of the computational resources used in the process, thereby promoting the responsible and sustainable progress of data science. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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Graphical abstract
Full article ">Figure 1
<p>Geographic locations of the studied sites (This map was generated using the Python package plotly.io).</p> Full article ">Figure 2
<p>Distribution of power generation.</p> Full article ">Figure 3
<p>Correlation matrix of all numerical variables within the database.</p> Full article ">Figure 4
<p>Correlation of all numerical variables in the database with the variable PolyPwr.</p> Full article ">Figure 5
<p>Comparative graph of the models proposed by Pasion et al.</p> Full article ">Figure 6
<p>Results obtained from location modelling using the parametrisation proposed by Pasion et al.</p> Full article ">Figure 7
<p>Comparison between results obtained from modelling using the parametrisation proposed by this study and by Pasion et al.</p> Full article ">Figure 8
<p>Results obtained from location modelling using the parametrisation proposed by this study.</p> Full article ">Figure 9
<p>Results from location modelling excluding a location using our proposed model.</p> Full article ">Figure 10
<p>Results from location modelling excluding a location and providing 1-week data context using our proposed model.</p> Full article ">
22 pages, 6759 KiB  
Article
Deep Learning Enriched Automation in Damage Detection for Sustainable Operation in Pipelines with Welding Defects under Varying Embedment Conditions
by Li Shang, Zi Zhang, Fujian Tang, Qi Cao, Nita Yodo, Hong Pan and Zhibin Lin
Computation 2023, 11(11), 218; https://doi.org/10.3390/computation11110218 - 2 Nov 2023
Cited by 6 | Viewed by 2035
Abstract
Welded joints in metallic pipelines and other structures are used to connect metallic structures. Welding defects, such as cracks and lack of fusion, are vulnerable to initiating early-age cracking and corrosion. The present damage identification techniques use ultrasonic-guided wave procedures, which depend on [...] Read more.
Welded joints in metallic pipelines and other structures are used to connect metallic structures. Welding defects, such as cracks and lack of fusion, are vulnerable to initiating early-age cracking and corrosion. The present damage identification techniques use ultrasonic-guided wave procedures, which depend on the change in the physical characteristics of waveforms as they propagate to determine damage states. However, the complexity of geometry and material discontinuity (e.g., the roughness of a weldment with or without defects) could lead to complicated wave reflection and scatters, thus increasing the difficulty in the signal processing. Artificial intelligence and machine learning exhibit their capability for data fusion, including processing signals originally from ultrasonic-guided waves. This study aims to utilize deep learning approaches, including a convolutional neural network (CNN), Long-short term memory network (LSTM), or hybrid CNN-LSTM model, to demonstrate the capability in automation for damage detection for pipes with welded joints embedded in soil. The damage features in terms of welding defect types and severity as well as multiple defects are used to understand the effectiveness of the hybrid CNN-LSTM model, which is further compared to the two commonly used deep learning approaches, CNN and LSTM. The results showed the hybrid CNN-LSTM model has much higher classification accuracy for damage states under all scenarios in comparison with the CNN and LSTM models. Furthermore, the impacts of the pipelines embedded in different types of materials, ranging from loose sand to stiff soil, on signal processing and data classification were further calibrated. The results demonstrated these deep learning approaches can still perform well to detect various pipeline damage under varying embedment conditions. However, the results demonstrate when concrete is used as an embedding material, high attention to absorbing the signal energy of concrete could pose a challenge for the signal processing, particularly under high noise levels. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Schematic diagram of the research methodology.</p> Full article ">Figure 2
<p>The structure of the CNN-LSTM hybrid model.</p> Full article ">Figure 3
<p>COMSOL model of pipeline under soil embedment.</p> Full article ">Figure 4
<p>Four kinds of welding defects.</p> Full article ">Figure 5
<p>Excited guided wave.</p> Full article ">Figure 6
<p>Pipeline waveforms under soil embedment.</p> Full article ">Figure 7
<p>The signals with varying degrees of noise disturbance.</p> Full article ">Figure 7 Cont.
<p>The signals with varying degrees of noise disturbance.</p> Full article ">Figure 8
<p>Accuracy of three deep learning models with different features.</p> Full article ">Figure 9
<p>The confusion matrix of the CNN-LSTM model with time- and frequency-domain features on different noise levels.</p> Full article ">Figure 10
<p>The confusion matrix showing the CNN-LSTM model with different noise levels.</p> Full article ">Figure 10 Cont.
<p>The confusion matrix showing the CNN-LSTM model with different noise levels.</p> Full article ">Figure 11
<p>ROC curve for three models on different noise levels.</p> Full article ">Figure 12
<p>Pipeline waveforms of four kind of defects with 10% severity under different kinds of embedment (Case 1).</p> Full article ">Figure 12 Cont.
<p>Pipeline waveforms of four kind of defects with 10% severity under different kinds of embedment (Case 1).</p> Full article ">Figure 13
<p>Accuracy of three deep learning models with different pipeline embedment.</p> Full article ">Figure 13 Cont.
<p>Accuracy of three deep learning models with different pipeline embedment.</p> Full article ">
28 pages, 9378 KiB  
Article
A Simulated-Annealing-Quasi-Oppositional-Teaching-Learning-Based Optimization Algorithm for Distributed Generation Allocation
by Seyed Iman Taheri, Mohammadreza Davoodi and Mohd. Hasan Ali
Computation 2023, 11(11), 214; https://doi.org/10.3390/computation11110214 - 2 Nov 2023
Cited by 2 | Viewed by 2440
Abstract
Conventional evolutionary optimization techniques often struggle with finding global optima, getting stuck in local optima instead, and can be sensitive to initial conditions and parameter settings. Efficient Distributed Generation (DG) allocation in distribution systems hinges on streamlined optimization algorithms that handle complex energy [...] Read more.
Conventional evolutionary optimization techniques often struggle with finding global optima, getting stuck in local optima instead, and can be sensitive to initial conditions and parameter settings. Efficient Distributed Generation (DG) allocation in distribution systems hinges on streamlined optimization algorithms that handle complex energy operations, support real-time decisions, adapt to dynamics, and improve system performance, considering cost and power quality. This paper proposes the Simulated-Annealing-Quasi-Oppositional-Teaching-Learning-Based Optimization Algorithm to efficiently allocate DGs within a distribution test system. The study focuses on wind turbines, photovoltaic units, and fuel cells as prominent DG due to their growing usage trends. The optimization goals include minimizing voltage losses, reducing costs, and mitigating greenhouse gas emissions in the distribution system. The proposed algorithm is implemented and evaluated on the IEEE 70-bus test system, with a comparative analysis conducted against other evolutionary methods such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Honey Bee Mating Optimization (HBMO), and Teaching-Learning-Based Optimization (TLBO) algorithms. Results indicate that the proposed algorithm is effective in allocating the DGs. Statistical testing confirms significant results (probability < 0.1), indicating superior optimization capabilities for this specific problem. Crucially, the proposed algorithm excels in both accuracy and computational speed compared to other methods studied. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Flowchart of the proposed optimization algorithm.</p> Full article ">Figure 2
<p>Location of 12 Fuel cell units with different objective functions by the single-objective MQOTLBO algorithm in the 70-bus distribution system.</p> Full article ">Figure 3
<p>MQOTLBO convergence curves for (<b>A</b>) the cost, (<b>B</b>) emission, (<b>C</b>) losses, and (<b>D</b>) voltage deviation.</p> Full article ">Figure 4
<p>Pareto front obtained using the MQTLBO algorithm.</p> Full article ">Figure 5
<p>Comparison of various methods by their obtained Pareto fronts.</p> Full article ">Figure 6
<p>Pareto front of three objective functions.</p> Full article ">Figure A1
<p>The illustration of membership functions for the objective function.</p> Full article ">Figure A2
<p>Learners score distribution (<b>a</b>) for classes 1 and 2 (<b>b</b>) for classes A and B.</p> Full article ">Figure A3
<p>Test system.</p> Full article ">
17 pages, 4411 KiB  
Article
Modeling of Wind Turbine Interactions and Wind Farm Losses Using the Velocity-Dependent Actuator Disc Model
by Ziemowit Malecha and Gideon Dsouza
Computation 2023, 11(11), 213; https://doi.org/10.3390/computation11110213 - 1 Nov 2023
Cited by 3 | Viewed by 2068
Abstract
This paper analyzes the interaction of wind turbines and losses in wind farms using computational fluid dynamics (CFD). The mathematical model used consisted of three-dimensional Reynolds-averaged Navier–Stokes (RANS) equations, while the presence of wind turbines in the flow was simulated as additional source [...] Read more.
This paper analyzes the interaction of wind turbines and losses in wind farms using computational fluid dynamics (CFD). The mathematical model used consisted of three-dimensional Reynolds-averaged Navier–Stokes (RANS) equations, while the presence of wind turbines in the flow was simulated as additional source terms. The novelty of the research is the definition of the source term as a velocity-dependent actuator disc model (ADM). This allowed for modeling the operation of a wind farm consisting of real wind turbines, characterized by power coefficients Cp and thrust force coefficients CT, which are a function of atmospheric wind speed. The calculations presented used a real 5 MW Gamesa turbine. Two different turbine spacings, 5D and 10D, where D is the diameter of the turbine, and two different locations corresponding to the offshore and onshore conditions were examined. The proposed model can be used to analyze wind farm losses not only in terms of the geometric distribution of individual turbines but also in terms of a specific type of wind turbine and in the entire wind speed spectrum. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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Full article ">Figure 1
<p>Sketch of the actuator disc model concept. Numbers 1, 2, 3, and 4 show the locations of the characteristic cross-sections used in linear momentum theory.</p> Full article ">Figure 2
<p>Actuator discs simulating two wind turbines placed one behind the other over rugged terrain. The characteristics of the turbines are described by the Equations (<a href="#FD5-computation-11-00213" class="html-disp-formula">5</a>) and (<a href="#FD6-computation-11-00213" class="html-disp-formula">6</a>). Calculation example taken from [<a href="#B42-computation-11-00213" class="html-bibr">42</a>,<a href="#B43-computation-11-00213" class="html-bibr">43</a>].</p> Full article ">Figure 3
<p>Power curve <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mrow> <mi>W</mi> <mi>T</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>T</mi> </msub> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of the <span class="html-italic">Gamesa G132-5.0MW</span> turbine considered in the current study [<a href="#B47-computation-11-00213" class="html-bibr">47</a>].</p> Full article ">Figure 4
<p>Front view (<math display="inline"><semantics> <mrow> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> </semantics></math> plane) of the placement of the actuator disc, dimensions, and boundaries.</p> Full article ">Figure 5
<p>Side view (<math display="inline"><semantics> <mrow> <mi>x</mi> <mo>,</mo> <mi>z</mi> </mrow> </semantics></math> plane) of the placement of the actuator discs, dimensions, and boundaries.</p> Full article ">Figure 6
<p>Numerical mesh detail with visibility of three levels of refinement. The actuator discs are located in the center of the region with the finest mesh.</p> Full article ">Figure 7
<p>Kinematic pressure distribution in the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math> plane at <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mi>W</mi> <mo>+</mo> <mn>0.5</mn> <mi>D</mi> </mrow> </semantics></math> for WF number 1.</p> Full article ">Figure 8
<p>Velocity magnitude distribution in the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math> plane at <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mi>W</mi> <mo>+</mo> <mn>0.5</mn> <mi>D</mi> </mrow> </semantics></math> for WF number 1.</p> Full article ">Figure 9
<p>Kinematic pressure and <span class="html-italic">x</span> component of velocity distribution along a line parallel to the <span class="html-italic">x</span> axis and passing through the center of the discs for WF number 1.</p> Full article ">Figure 10
<p>Comparison of the velocity profiles in the wake between the single-column wind farm WF1 and multi-column wind farm WF2. The plots are taken in the direction of the width direction <span class="html-italic">y</span> just behind the actuator discs and at the height <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>120</mn> </mrow> </semantics></math> m (the hub height).</p> Full article ">Figure 11
<p>Comparison of <math display="inline"><semantics> <msub> <mi>U</mi> <mi>x</mi> </msub> </semantics></math> velocity between the corresponding onshore and offshore wind farms (see <a href="#computation-11-00213-t002" class="html-table">Table 2</a>). Plots are taken along the line that goes through the centers of the actuator discs.</p> Full article ">Figure 12
<p>Weibull wind velocity probability distribution for the onshore location and modified wind velocity distributions reaching the turbines in subsequent rows. Data for WF1.</p> Full article ">
12 pages, 5127 KiB  
Article
Transformer-Based Model for Predicting Customers’ Next Purchase Day in e-Commerce
by Alexandru Grigoraș and Florin Leon
Computation 2023, 11(11), 210; https://doi.org/10.3390/computation11110210 - 29 Oct 2023
Cited by 2 | Viewed by 4516
Abstract
The paper focuses on predicting the next purchase day (NPD) for customers in e-commerce, a task with applications in marketing, inventory management, and customer retention. A novel transformer-based model for NPD prediction is introduced and compared to traditional methods such as ARIMA, XGBoost, [...] Read more.
The paper focuses on predicting the next purchase day (NPD) for customers in e-commerce, a task with applications in marketing, inventory management, and customer retention. A novel transformer-based model for NPD prediction is introduced and compared to traditional methods such as ARIMA, XGBoost, and LSTM. Transformers offer advantages in capturing long-term dependencies within time series data through self-attention mechanisms. This adaptability to various time series patterns, including trends, seasonality, and irregularities, makes them a promising choice for NPD prediction. The transformer model demonstrates improvements in prediction accuracy compared to the baselines. Additionally, a clustered transformer model is proposed, which further enhances accuracy, emphasizing the potential of this architecture for NPD prediction. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Days between orders feature generation (adapted after [<a href="#B5-computation-11-00210" class="html-bibr">5</a>]).</p> Full article ">Figure 2
<p>Converting the time series data into a supervised learning problem (adapted after [<a href="#B18-computation-11-00210" class="html-bibr">18</a>]).</p> Full article ">Figure 3
<p>The proposed transformer architecture (adapted from [<a href="#B14-computation-11-00210" class="html-bibr">14</a>]).</p> Full article ">Figure 4
<p>Customers’ recency distribution.</p> Full article ">Figure 5
<p>The frequency distribution of customers’ orders.</p> Full article ">Figure 6
<p>The within clusters sum of squared errors (SSE) employed to select the most appropriate number of clusters using the elbow method.</p> Full article ">Figure 7
<p>Time series data with the NPDs for customer 14395.</p> Full article ">Figure 8
<p>Time series data with next purchase days for customer 14911.</p> Full article ">Figure 9
<p>Time series using selected models for customer 14395.</p> Full article ">Figure 10
<p>Time series using selected models for customer 14911.</p> Full article ">Figure 11
<p>The errors obtained by the selected models.</p> Full article ">
16 pages, 465 KiB  
Article
Numerical Solution of the Retrospective Inverse Parabolic Problem on Disjoint Intervals
by Miglena N. Koleva and Lubin G. Vulkov
Computation 2023, 11(10), 204; https://doi.org/10.3390/computation11100204 - 16 Oct 2023
Cited by 1 | Viewed by 1543
Abstract
The retrospective inverse problem for evolution equations is formulated as the reconstruction of unknown initial data by a given solution at the final time. We consider the inverse retrospective problem for a one-dimensional parabolic equation in two disconnected intervals with weak solutions in [...] Read more.
The retrospective inverse problem for evolution equations is formulated as the reconstruction of unknown initial data by a given solution at the final time. We consider the inverse retrospective problem for a one-dimensional parabolic equation in two disconnected intervals with weak solutions in weighted Sobolev spaces. The two solutions are connected with nonstandard interface conditions, and thus this problem is solved in the whole spatial region. Such a problem, as with other inverse problems, is ill-posed, and for its numerical solution, specific techniques have to be used. The direct problem is first discretized by a difference scheme which provides a second order of approximation in space. For the resulting ordinary differential equation system, the positive coerciveness is established. Next, we develop an iterative conjugate gradient method to solve the ill-posed systems of the difference equations, which are obtained after weighted time discretization, of the inverse problem. Test examples with noisy input data are discussed. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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Full article ">Figure 1
<p>Exact solution <math display="inline"><semantics> <mfenced separators="" open="(" close=")"> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </mfenced> </semantics></math> (solid line) and numerical solution <math display="inline"><semantics> <mfenced separators="" open="(" close=")"> <msubsup> <mi>u</mi> <mrow> <mn>1</mn> <mo>,</mo> <msub> <mi>i</mi> <mn>1</mn> </msub> </mrow> <mi>T</mi> </msubsup> <mo>,</mo> <msubsup> <mi>u</mi> <mrow> <mn>2</mn> <mo>,</mo> <msub> <mi>i</mi> <mn>2</mn> </msub> </mrow> <mi>T</mi> </msubsup> </mfenced> </semantics></math> (line with circles) of the direct problem at the final time (<span class="html-italic"><b>left</b></span>) and the corresponding error (<span class="html-italic"><b>right</b></span>), where <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mi>h</mi> </mrow> </semantics></math> (Example 1).</p> Full article ">Figure 2
<p>Exact solution <math display="inline"><semantics> <mfenced separators="" open="(" close=")"> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </mfenced> </semantics></math> (solid line) and numerical solution to the direct problem <math display="inline"><semantics> <mfenced separators="" open="(" close=")"> <msubsup> <mi>u</mi> <mrow> <mn>1</mn> <mo>,</mo> <msub> <mi>i</mi> <mn>1</mn> </msub> </mrow> <mi>T</mi> </msubsup> <mo>,</mo> <msubsup> <mi>u</mi> <mrow> <mn>2</mn> <mo>,</mo> <msub> <mi>i</mi> <mn>2</mn> </msub> </mrow> <mi>T</mi> </msubsup> </mfenced> </semantics></math> (line with circles) at the final time (<span class="html-italic"><b>left</b></span>) and the corresponding error (<span class="html-italic"><b>right</b></span>), where <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mi>h</mi> </mrow> </semantics></math> (Example 1).</p> Full article ">Figure 3
<p>True (solid line) and recovered (line with circles) initial functions (<span class="html-italic"><b>left</b></span>) and the corresponding error (<span class="html-italic"><b>right</b></span>), where <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mi>h</mi> </mrow> </semantics></math> (Example 2).</p> Full article ">Figure 4
<p>True (solid line) and recovered (line with circles) initial functions (<span class="html-italic"><b>left</b></span>) and the corresponding error (<span class="html-italic"><b>right</b></span>), where <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mi>h</mi> </mrow> </semantics></math> (Example 2).</p> Full article ">Figure 5
<p>True (solid line) and recovered (line with circles) solutions <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> (<span class="html-italic"><b>left</b></span>) and the corresponding error (<span class="html-italic"><b>right</b></span>), where <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mi>h</mi> </mrow> </semantics></math> (Example 2).</p> Full article ">Figure 6
<p>True (solid line) and recovered (line with circles) solutions <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> (<span class="html-italic"><b>left</b></span>) and the corresponding error (<span class="html-italic"><b>right</b></span>), where <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mi>h</mi> </mrow> </semantics></math> (Example 2).</p> Full article ">Figure 7
<p>True (solid line) and recovered (line with circles) initial functions (<span class="html-italic"><b>left</b></span>) and the corresponding error (<span class="html-italic"><b>right</b></span>), where <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mi>h</mi> </mrow> </semantics></math> with smoothed measurements (Example 2).</p> Full article ">Figure 8
<p>True (solid line) and recovered (line with circles) solutions <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> (<span class="html-italic"><b>left</b></span>) and the corresponding error (<span class="html-italic"><b>right</b></span>), where <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mi>h</mi> </mrow> </semantics></math> with smoothed measurements (Example 2).</p> Full article ">
12 pages, 2940 KiB  
Article
A Graphical Calibration Method for a Water Quality Model Considering Process Variability Versus Delay Time: Theory and a Case Study
by Eyal Brill and Michael Bendersky
Computation 2023, 11(10), 200; https://doi.org/10.3390/computation11100200 - 7 Oct 2023
Viewed by 1471
Abstract
Process Variability (PV) is a significant water quality time-series measurement. It is a critical element in detecting abnormality. Typically, the quality control system should raise an alert if the PV exceeds its normal value after a proper delay time (DT). The literature does [...] Read more.
Process Variability (PV) is a significant water quality time-series measurement. It is a critical element in detecting abnormality. Typically, the quality control system should raise an alert if the PV exceeds its normal value after a proper delay time (DT). The literature does not address the relation between the extended process variability and the time delay for a warning. The current paper shows a graphical method for calibrating a Water Quality Model based on these two parameters. The amount of variability is calculated based on the Euclidean distance between records in a dataset. Typically, each multivariable process has some relation between the variability and the time delay. In the case of a short period (a few minutes), the PV may be high. However, as the relevant DT is longer, it is expected to see the PV converge to some steady state. The current paper examines a method for estimating the relationship between the two measurements (PV and DT) as a detection tool for abnormality. Given the user’s classification of the actual event for true and false events, the method shows how to build a graphical map that helps the user select the best thresholds for the model. The last section of the paper offers an implementation of the method using real-world data. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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Figure 1

Figure 1
<p>Actual Traveling Distance (ATD) of the virtual center point (VCP).</p> Full article ">Figure 2
<p>Relation between the Distance and Delay Time.</p> Full article ">Figure 3
<p>PV vs. DT for different S curves.</p> Full article ">Figure 4
<p>Events over decision thresholds.</p> Full article ">Figure 5
<p>Raw data water quality chart for Chlorine (mg/mL).</p> Full article ">Figure 6
<p>Raw data water quality chart for Conductivity (micro-Siemens).</p> Full article ">Figure 7
<p>Raw data water quality chart for pH.</p> Full article ">Figure 8
<p>Raw data water quality chart for Turbidity.</p> Full article ">Figure 9
<p>Raw data water quality chart for Temperature.</p> Full article ">Figure 10
<p>Process variability (PV) values using specific LAG and Threshold.</p> Full article ">
19 pages, 4282 KiB  
Article
Fuzzy Transform Image Compression in the YUV Space
by Barbara Cardone, Ferdinando Di Martino and Salvatore Sessa
Computation 2023, 11(10), 191; https://doi.org/10.3390/computation11100191 - 1 Oct 2023
Viewed by 1795
Abstract
This research proposes a new image compression method based on the F1-transform which improves the quality of the reconstructed image without increasing the coding/decoding CPU time. The advantage of compressing color images in the YUV space is due to the fact that while [...] Read more.
This research proposes a new image compression method based on the F1-transform which improves the quality of the reconstructed image without increasing the coding/decoding CPU time. The advantage of compressing color images in the YUV space is due to the fact that while the three bands Red, Green and Blue are equally perceived by the human eye, in YUV space most of the image information perceived by the human eye is contained in the Y band, as opposed to the U and V bands. Using this advantage, we construct a new color image compression algorithm based on F1-transform in which the image compression is accomplished in the YUV space, so that better-quality compressed images can be obtained without increasing the execution time. The results of tests performed on a set of color images show that our color image compression method improves the quality of the decoded images with respect to the image compression algorithms JPEG, F1-transform on the RGB color space and F-transform on the YUV color space, regardless of the selected compression rate and with comparable CPU times. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
►▼ Show Figures

Figure 1

Figure 1
<p>Flow diagram of the YUV F<sup>1</sup>-transform image compression algorithm.</p> Full article ">Figure 2
<p>Flow diagram of the YUV F1-transform image decompression algorithm.</p> Full article ">Figure 3
<p>Source images: (<b>a</b>) 256 × 256 image 4.1.04; (<b>b</b>): 512 × 512 image 4.2.07.</p> Full article ">Figure 4
<p>Decoded image 4.1.04, ρ ≈ 0.10, obtained via: (<b>a</b>) JPEG; (<b>b</b>) F1-transform; (<b>c</b>): YUV F-transform; (<b>d</b>) YUV F1-transform.</p> Full article ">Figure 5
<p>Decoded image 4.1.04, ρ ≈ 0.25, obtained via: (<b>a</b>) JPEG; (<b>b</b>) F1-transform; (<b>c</b>): YUV F-transform; (<b>d</b>) YUV F1-transform.</p> Full article ">Figure 6
<p>PSNR trend for the color image 4.1.04 obtained by executing the four color image compressions algorithms.</p> Full article ">Figure 7
<p>Decoded image 4.2.07, ρ ≈ 0.10, obtained via: (<b>a</b>) JPEG; (<b>b</b>) F1-transform; (<b>c</b>): YUV F-transform; (<b>d</b>) YUV F1-transform.</p> Full article ">Figure 8
<p>Decoded image 4.2.07, ρ ≈ 0.25, obtained via: (<b>a</b>) JPEG; (<b>b</b>) F1-transform; (<b>c</b>): YUV F-transform; (<b>d</b>) YUV F1-transform.</p> Full article ">Figure 9
<p>PSNR trend for the color image 4.2.07 obtained by executing the four color image compression algorithms.</p> Full article ">Figure 10
<p>Trend of the Gain of YUV-F1transform with respect to the other three color image compression methods.</p> Full article ">
20 pages, 4220 KiB  
Article
Estimation of Temperature-Dependent Thermal Conductivity and Heat Capacity Given Boundary Data
by Abdulaziz Sharahy and Zaid Sawlan
Computation 2023, 11(9), 184; https://doi.org/10.3390/computation11090184 - 14 Sep 2023
Cited by 2 | Viewed by 1793
Abstract
This work aims to estimate temperature-dependent thermal conductivity and heat capacity given measurements of temperature and heat flux at the boundaries. This estimation problem has many engineering and industrial applications, such as those for the building sector and chemical reactors. Two approaches are [...] Read more.
This work aims to estimate temperature-dependent thermal conductivity and heat capacity given measurements of temperature and heat flux at the boundaries. This estimation problem has many engineering and industrial applications, such as those for the building sector and chemical reactors. Two approaches are proposed to address this problem. The first method uses an integral approach and a polynomial approximation of the temperature profile. The second method uses a numerical solver for the nonlinear heat equation and an optimization algorithm. The performance of the two methods is compared using synthetic data generated with different boundary conditions and configurations. The results demonstrate that the integral approach works in limited scenarios, whereas the numerical approach is effective in estimating temperature-dependent thermal properties. The second method is also extended to account for noisy measurements and a comprehensive uncertainty quantification framework is developed. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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Figure 1

Figure 1
<p>Plot of the temperature and heat flux at the boundary in Dataset 1.</p> Full article ">Figure 2
<p>Different plots of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (left side) and <math display="inline"><semantics> <mrow> <msubsup> <mi>f</mi> <mn>2</mn> <mi>N</mi> </msubsup> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (right side) using Dataset 1. (<b>a</b>) Plot of <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and varying <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>b</mi> <mi>c</mi> </msub> </semantics></math>. (<b>b</b>) Plot of <math display="inline"><semantics> <msubsup> <mi>f</mi> <mn>2</mn> <mi>N</mi> </msubsup> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and varying <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>b</mi> <mi>c</mi> </msub> </semantics></math>. (<b>c</b>) Plot of <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and varying <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>a</mi> <mi>k</mi> </msub> </semantics></math>. (<b>d</b>) Plot of <math display="inline"><semantics> <msubsup> <mi>f</mi> <mn>2</mn> <mi>N</mi> </msubsup> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and varying <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>a</mi> <mi>k</mi> </msub> </semantics></math>. (<b>e</b>) Plot of <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and varying <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>b</mi> <mi>k</mi> </msub> </semantics></math>. (<b>f</b>) Plot of <math display="inline"><semantics> <msubsup> <mi>f</mi> <mn>2</mn> <mi>N</mi> </msubsup> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and varying <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>b</mi> <mi>k</mi> </msub> </semantics></math>.</p> Full article ">Figure 2 Cont.
<p>Different plots of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (left side) and <math display="inline"><semantics> <mrow> <msubsup> <mi>f</mi> <mn>2</mn> <mi>N</mi> </msubsup> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (right side) using Dataset 1. (<b>a</b>) Plot of <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and varying <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>b</mi> <mi>c</mi> </msub> </semantics></math>. (<b>b</b>) Plot of <math display="inline"><semantics> <msubsup> <mi>f</mi> <mn>2</mn> <mi>N</mi> </msubsup> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>3</mn> <mo>,</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and varying <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>b</mi> <mi>c</mi> </msub> </semantics></math>. (<b>c</b>) Plot of <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and varying <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>a</mi> <mi>k</mi> </msub> </semantics></math>. (<b>d</b>) Plot of <math display="inline"><semantics> <msubsup> <mi>f</mi> <mn>2</mn> <mi>N</mi> </msubsup> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and varying <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>a</mi> <mi>k</mi> </msub> </semantics></math>. (<b>e</b>) Plot of <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and varying <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>b</mi> <mi>k</mi> </msub> </semantics></math>. (<b>f</b>) Plot of <math display="inline"><semantics> <msubsup> <mi>f</mi> <mn>2</mn> <mi>N</mi> </msubsup> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and varying <math display="inline"><semantics> <msub> <mi>a</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>b</mi> <mi>k</mi> </msub> </semantics></math>.</p> Full article ">Figure 3
<p>Plot of the temperature and heat flux at the boundary in Dataset 2.</p> Full article ">Figure 4
<p>Different plots of <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> using Dataset 2 with <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>}</mo> </mrow> </mrow> </semantics></math>. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Different plots of <math display="inline"><semantics> <msubsup> <mi>f</mi> <mn>2</mn> <mi>N</mi> </msubsup> </semantics></math> using Dataset 2 with <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>}</mo> </mrow> </mrow> </semantics></math>. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Plot of the temperature and heat flux at the boundary in Dataset 3.</p> Full article ">Figure 7
<p>Plots of <math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msubsup> <mi>f</mi> <mn>2</mn> <mi>N</mi> </msubsup> </semantics></math> using Dataset 3 with <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mi>c</mi> </msub> <mo>=</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Plot of one estimated sample of the boundary conditions with the noisy measurements.</p> Full article ">Figure 9
<p>Scatter plots of the estimated samples of <math display="inline"><semantics> <mi>θ</mi> </semantics></math> using noisy Dataset 1.</p> Full article ">Figure 10
<p>Plot of one estimated sample of the boundary conditions with the noisy measurements.</p> Full article ">Figure 11
<p>Scatter plots of the estimated samples of <math display="inline"><semantics> <mi>θ</mi> </semantics></math> using noisy Dataset 2.</p> Full article ">
28 pages, 13353 KiB  
Article
In-Silico Prediction of Mechanical Behaviour of Uniform Gyroid Scaffolds Affected by Its Design Parameters for Bone Tissue Engineering Applications
by Haja-Sherief N. Musthafa, Jason Walker, Talal Rahman, Alvhild Bjørkum, Kamal Mustafa and Dhayalan Velauthapillai
Computation 2023, 11(9), 181; https://doi.org/10.3390/computation11090181 - 12 Sep 2023
Cited by 4 | Viewed by 2591
Abstract
Due to their excellent properties, triply periodic minimal surfaces (TPMS) have been applied to design scaffolds for bone tissue engineering applications. Predicting the mechanical response of bone scaffolds in different loading conditions is vital to designing scaffolds. The optimal mechanical properties can be [...] Read more.
Due to their excellent properties, triply periodic minimal surfaces (TPMS) have been applied to design scaffolds for bone tissue engineering applications. Predicting the mechanical response of bone scaffolds in different loading conditions is vital to designing scaffolds. The optimal mechanical properties can be achieved by tuning their geometrical parameters to mimic the mechanical properties of natural bone. In this study, we designed gyroid scaffolds of different user-specific pore and strut sizes using a combined TPMS and signed distance field (SDF) method to obtain varying architecture and porosities. The designed scaffolds were converted to various meshes such as surface, volume, and finite element (FE) volume meshes to create FE models with different boundary and loading conditions. The designed scaffolds under compressive loading were numerically evaluated using a finite element method (FEM) to predict and compare effective elastic moduli. The effective elastic moduli range from 0.05 GPa to 1.93 GPa was predicted for scaffolds of different architectures comparable to human trabecular bone. The results assert that the optimal mechanical properties of the scaffolds can be achieved by tuning their design and morphological parameters to match the mechanical properties of human bone. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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Figure 1

Figure 1
<p>(<b>a</b>) A triply periodic minimal surface-based gyroid scaffold of pore size 1000 µm and strut size of 200 µm with a well-interconnected network of pores and struts helping the movement of oxygen, nutrients, and waste materials. (<b>b</b>) Illustration of how relative density affects volume fraction and influences the morphology of a unit cell. (<b>c</b>) Representative of a gyroid unit cell’s pore size (red) and strut size (black) [<a href="#B6-computation-11-00181" class="html-bibr">6</a>].</p> Full article ">Figure 1 Cont.
<p>(<b>a</b>) A triply periodic minimal surface-based gyroid scaffold of pore size 1000 µm and strut size of 200 µm with a well-interconnected network of pores and struts helping the movement of oxygen, nutrients, and waste materials. (<b>b</b>) Illustration of how relative density affects volume fraction and influences the morphology of a unit cell. (<b>c</b>) Representative of a gyroid unit cell’s pore size (red) and strut size (black) [<a href="#B6-computation-11-00181" class="html-bibr">6</a>].</p> Full article ">Figure 2
<p>Workflow of the given research project for the design of scaffolds and FEM-based compressive loading simulation. The nTop notebooks for design of scaffolds (<a href="#app2-computation-11-00181" class="html-app">Appendix A</a> and <a href="#app5-computation-11-00181" class="html-app">Appendix D</a>), creation of FE volume mesh (<a href="#app4-computation-11-00181" class="html-app">Appendix C</a>) and static structural analysis (<a href="#app5-computation-11-00181" class="html-app">Appendix D</a>) can be seen.</p> Full article ">Figure 3
<p>An illustrative concept of an SDF in 2D.</p> Full article ">Figure 4
<p>The Boolean intersection between the gyroid implicit surface and the required shape (cuboid) of 5 × 5 × 10 mm<sup>3</sup> to obtain a cuboid gyroid scaffold of 5 × 5 × 10 mm<sup>3</sup>.</p> Full article ">Figure 5
<p>Uniform Gyroid Scaffolds of different pore sizes (PS) ranging from 200 µm to 1000 µm with a constant strut size (SS) of 200 µm.</p> Full article ">Figure 6
<p>Workflow of the conversion process of different meshes for simulation.</p> Full article ">Figure 7
<p>A surface mesh having triangular elements.</p> Full article ">Figure 8
<p>(<b>a</b>) A volume mesh having tetrahedral elements, which are 3D solid volume elements, different from a surface mesh of 2D elements; (<b>b</b>) A FE volume mesh having integration points of a geometric order added to a volume mesh.</p> Full article ">Figure 9
<p>Convergence plots of PS350 for (<b>a</b>) displacement and (<b>b</b>) Max. von Mises Stress (Remaining plots can be viewed in <a href="#app1-computation-11-00181" class="html-app">Supplementary Materials Figure S1</a>).</p> Full article ">Figure 10
<p>Frontal and isometric views of an FE model, which is a combination of FE volume mesh with boundary conditions—a uniform force (Yellow) is applied on a top plate (movable), and a displacement restraint (red) is applied on a bottom plate (fixed).</p> Full article ">Figure 11
<p>(<b>a</b>) Reactive force. (<b>b</b>) Equivalent area of a scaffold.</p> Full article ">Figure 12
<p>Graph of Design and Morphological Properties: variations of (<b>a</b>) surface area with poresize, (<b>b</b>) level constant with pore / strut ratio, (<b>c</b>) period coefficient with level constant, (<b>d</b>) surface area to volume ration with pore size, (<b>e</b>) porosity with pore size, (<b>f</b>) level constant with pore size, (<b>g</b>) volume fraction with level constant, and (<b>h</b>) volume fraction with pore size.</p> Full article ">Figure 13
<p>FE models under compressive loading—the von Mises distribution of gyroid scaffold PS350. The stress values increment from violet (minimum value) to red colour (maximum value). The von Mises contours of other FE models can be seen in <a href="#app1-computation-11-00181" class="html-app">Supplementary Materials Table S2</a>.</p> Full article ">Figure 14
<p>FE models under compressive loading. Displacement distribution of gyroid scaffold PS350. The displacement values increment from violet (minimum value) to red colour (maximum value). The displacement contours of other FE models can be seen in <a href="#app1-computation-11-00181" class="html-app">Supplementary Materials Table S3</a>.</p> Full article ">Figure 15
<p>Graph of mechanical properties predicted from FE simulation: variations of effective elastic modulus with (<b>a</b>) pore size, (<b>b</b>) porosity, and (<b>c</b>) volume fraction; variations of (<b>d</b>) relative elastic modulus with volume fraction, (<b>e</b>) maximum von Mises stress with maximum deformation and (<b>f</b>) maximum von Mises stress with pore size.</p> Full article ">Figure A1
<p>Front Panel of Virtual Instrument (vi) for calculating C and N.</p> Full article ">Figure A2
<p>Block diagrams of vi.</p> Full article ">Figure A3
<p>Design of a gyroid scaffold PS350 in nTopology.</p> Full article ">Figure A4
<p>Conversion of Femur shape into PS350 gyroid architecture.</p> Full article ">Figure A5
<p>Different stages of meshing from surface mesh to FE volume mesh.</p> Full article ">Figure A6
<p>A FE model of PS350 and its related static structural analysis.</p> Full article ">
11 pages, 1999 KiB  
Article
Solving the Problem of Elasticity for a Layer with N Cylindrical Embedded Supports
by Vitaly Miroshnikov, Oleksandr Savin, Vladimir Sobol and Vyacheslav Nikichanov
Computation 2023, 11(9), 172; https://doi.org/10.3390/computation11090172 - 3 Sep 2023
Cited by 2 | Viewed by 1264
Abstract
The main goal of deformable solid mechanics is to determine the stress–strain state of parts, structural elements, and their connections. The most accurate results of calculations of this state allow us to optimize design objects. However, not all models can be solved using [...] Read more.
The main goal of deformable solid mechanics is to determine the stress–strain state of parts, structural elements, and their connections. The most accurate results of calculations of this state allow us to optimize design objects. However, not all models can be solved using exact methods. One such model is the problem of a layer with cylindrical embedded supports that are parallel to each other and the layer boundaries. In this work, the supports are represented by cylindrical cavities with zero displacements set on them. The layer is considered in Cartesian coordinates, and the cavities are in cylindrical coordinates. To solve the problem, the Lamé equation is used, where the basic solutions between different coordinate systems are linked using the generalized Fourier method. By satisfying the boundary conditions and linking different coordinate systems, a system of infinite linear algebraic equations is created. For numerical realization, the method of reduction is used to find the unknowns. The numerical analysis has shown that the boundary conditions are fulfilled with high accuracy, and the physical pattern of the stress distribution and the comparison with results of similar studies indicate the accuracy of the obtained results. The proposed method for calculating the stress–strain state can be applied to the calculation of structures whose model is a layer with cylindrical embedded supports. The numerical results of the work make it possible to predetermine the geometric parameters of the model to be designed. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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Figure 1
<p>Layer with rigidly fixed cylindrical cavities.</p> Full article ">Figure 2
<p>Stress distribution on the surface of the cavity <span class="html-italic">p</span> = 1.</p> Full article ">Figure 3
<p>Stress <math display="inline"><semantics> <mrow> <msub> <mo>σ</mo> <mo>φ</mo> </msub> </mrow> </semantics></math> on the surface of the cavity <span class="html-italic">p</span> = 1.</p> Full article ">Figure 4
<p>Stress <math display="inline"><semantics> <mrow> <msub> <mo>τ</mo> <mrow> <mi mathvariant="sans-serif">ρ</mi> <mo>φ</mo> </mrow> </msub> </mrow> </semantics></math> on the surface of the cavity <span class="html-italic">p</span> = 1.</p> Full article ">Figure 5
<p>Stresses <math display="inline"><semantics> <mrow> <msub> <mo>σ</mo> <mo>φ</mo> </msub> </mrow> </semantics></math> on the cavity surfaces <span class="html-italic">p</span> = 1 and <span class="html-italic">p</span> = 2.</p> Full article ">Figure 6
<p>The stress <math display="inline"><semantics> <mrow> <msub> <mo>σ</mo> <mi>x</mi> </msub> </mrow> </semantics></math> at the top of the layer.</p> Full article ">
23 pages, 5619 KiB  
Article
Impact of Cross-Tie Material Nonlinearity on the Dynamic Behavior of Shallow Flexible Cable Networks
by Amir Younespour and Shaohong Cheng
Computation 2023, 11(9), 169; https://doi.org/10.3390/computation11090169 - 1 Sep 2023
Viewed by 1127
Abstract
Cross-ties have proven their efficacy in mitigating vibrations in bridge stay cables. Several factors, such as cross-tie malfunctions due to slackening or snapping, as well as the utilization of high-energy dissipative materials, can introduce nonlinear restoring forces in the cross-ties. While previous studies [...] Read more.
Cross-ties have proven their efficacy in mitigating vibrations in bridge stay cables. Several factors, such as cross-tie malfunctions due to slackening or snapping, as well as the utilization of high-energy dissipative materials, can introduce nonlinear restoring forces in the cross-ties. While previous studies have investigated the influence of the former on cable network dynamics, the evaluation of the impact of nonlinear cross-tie materials remains unexplored. In this current research, an existing analytical model of a two-shallow-flexible-cable network has been extended to incorporate the cross-tie material nonlinearity in the formulation. The harmonic balance method (HBM) is employed to determine the equivalent linear stiffness of the cross-ties. The dynamic response of a cable network containing nonlinear cross-ties is approximated by comparing it to an equivalent linear system. Additionally, the study delves into the effects of the cable vibration amplitude, cross-tie material properties, installation location, and the length ratio between constituent cables on both the fundamental frequency of the cable network and the equivalent linear stiffness of the cross-ties. The findings reveal that the presence of cross-tie nonlinearity significantly influences the in-plane modal response of the cable network. Not only the frequencies of all the modes are reduced, but the formation of local modes is delayed to a high order. In contrast to an earlier finding based on a linear cross-tie assumption, with nonlinearity present, moving a cross-tie towards the mid-span of a cable would not enhance the in-plane stiffness of the network. Moreover, the impact of the length ratio on the network in-plane stiffness and frequency is contingent on its combined effect on the cross-tie axial stiffness and the lateral stiffness of neighboring cables. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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<p>Cable cross-tie [<a href="#B27-computation-11-00169" class="html-bibr">27</a>].</p> Full article ">Figure 2
<p>Schematic diagram of the mathematical model of a two-shallow-flexible-cable network with a nonlinear cross-tie.</p> Full article ">Figure 3
<p>Fitted curve to the original non-dimensional force-displacement data of the cross-tie material (strain-softening behavior).</p> Full article ">Figure 4
<p>Comparison of the modal response of the first 10 modes of a twin-cable network with either a linear or a nonlinear cross-tie (GM: global mode; LM: local mode; Sym: symmetric; Asym: anti-symmetric).</p> Full article ">Figure 5
<p>Comparison of the modal response of first 10 modes of an unequal-length two-cable network with either a linear or a nonlinear cross-tie (GM: global mode; LM: local mode; PS: pseudo symmetric; PAS: pseudo anti-symmetric).</p> Full article ">Figure 6
<p>Fitted curve to the original non-dimensional force-displacement data of the cross-tie material (strain-hardening behavior).</p> Full article ">Figure 7
<p>Effect of the cross-tie material nonlinearity type on different system parameters (<math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>): (<b>a</b>) non-dimensional fundamental frequency of the cable network; (<b>b</b>) non-dimensional equivalent linear stiffness of the cross-tie.</p> Full article ">Figure 8
<p>Effect of the cross-tie position on the non-dimensional equivalent linear stiffness of the cross-tie and non-dimensional fundamental frequency of the cable network: (<b>a</b>) strain-softening cross-tie behavior; (<b>b</b>) strain-hardening cross-tie behavior; (<b>c</b>) relationship between the system fundamental frequency and cross-tie equivalent linear stiffness for strain-softening-type material; (<b>d</b>) relationship between the system fundamental frequency and cross-tie equivalent linear stiffness for strain-hardening-type material.</p> Full article ">Figure 9
<p>Effect of the cross-tie position on the degree of the cross-tie deformation for the first mode of vibration.</p> Full article ">Figure 10
<p>Effect of the length ratio on the cross-tie deformation in the fundamental mode of a two-cable network.</p> Full article ">Figure 11
<p>Effect of the length ratio on the non-dimensional equivalent linear stiffness <math display="inline"><semantics> <mrow> <msub> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>K</mi> </mrow> <mo>¯</mo> </mover> </mrow> <mrow> <mi>e</mi> <mi>q</mi> </mrow> </msub> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> of the cross-tie (<math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>): (<b>a</b>) strain-softening cross-tie behavior; (<b>b</b>) strain-hardening cross-tie behavior.</p> Full article ">Figure 12
<p>Effect of the length ratio on the non-dimensional fundamental frequency of a two-cable network (<math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>): (<b>a</b>) strain-softening cross-tie behavior; (<b>b</b>) strain-hardening cross-tie behavior.</p> Full article ">
27 pages, 1255 KiB  
Article
Adapting PINN Models of Physical Entities to Dynamical Data
by Dmitriy Tarkhov, Tatiana Lazovskaya and Valery Antonov
Computation 2023, 11(9), 168; https://doi.org/10.3390/computation11090168 - 1 Sep 2023
Viewed by 1823
Abstract
This article examines the possibilities of adapting approximate solutions of boundary value problems for differential equations using physics-informed neural networks (PINNs) to changes in data about the physical entity being modelled. Two types of models are considered: PINN and parametric PINN (PPINN). The [...] Read more.
This article examines the possibilities of adapting approximate solutions of boundary value problems for differential equations using physics-informed neural networks (PINNs) to changes in data about the physical entity being modelled. Two types of models are considered: PINN and parametric PINN (PPINN). The former is constructed for a fixed parameter of the problem, while the latter includes the parameter for the number of input variables. The models are tested on three problems. The first problem involves modelling the bending of a cantilever rod under varying loads. The second task is a non-stationary problem of a thermal explosion in the plane-parallel case. The initial model is constructed based on an ordinary differential equation, while the modelling object satisfies a partial differential equation. The third task is to solve a partial differential equation of mixed type depending on time. In all cases, the initial models are adapted to the corresponding pseudo-measurements generated based on changing equations. A series of experiments are carried out for each problem with different functions of a parameter that reflects the character of changes in the object. A comparative analysis of the quality of the PINN and PPINN models and their resistance to data changes has been conducted for the first time in this study. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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Figure 1
<p>Schematic of the PINN (<b>left</b>) and PPINN (<b>right</b>) for solving differential equations and adapting to measurements.</p> Full article ">Figure 2
<p>Bending of the Cantilever Rod under Load Scheme.</p> Full article ">Figure 3
<p>Different variants of the parameter dependence (<a href="#FD12-computation-11-00168" class="html-disp-formula">12</a>) and (<a href="#FD13-computation-11-00168" class="html-disp-formula">13</a>) on time.</p> Full article ">Figure 4
<p>Different variants of the parameter dependence (<a href="#FD16-computation-11-00168" class="html-disp-formula">16</a>)–(<a href="#FD19-computation-11-00168" class="html-disp-formula">19</a>) on time.</p> Full article ">Figure 5
<p>Different variants of the parameter dependence (<a href="#FD22-computation-11-00168" class="html-disp-formula">22</a>) and (<a href="#FD23-computation-11-00168" class="html-disp-formula">23</a>) on time.</p> Full article ">Figure 6
<p>Comparison of the exact solution <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math> to (<a href="#FD9-computation-11-00168" class="html-disp-formula">9</a>) and (<a href="#FD10-computation-11-00168" class="html-disp-formula">10</a>) and the approximate solutions <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="bold">c</mi> <mo>,</mo> <mi mathvariant="bold">a</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD33-computation-11-00168" class="html-disp-formula">33</a>) and <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mrow> <mi mathvariant="bold">c</mi> <mo>,</mo> <mi mathvariant="bold">a</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD34-computation-11-00168" class="html-disp-formula">34</a>) at parameter value <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (<b>left</b>) along with the absolute difference between the exact and approximate solutions at the same parameter value (<b>right</b>).</p> Full article ">Figure 7
<p>The absolute errors of PPINN (<a href="#FD29-computation-11-00168" class="html-disp-formula">29</a>) approximations <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="bold">c</mi> <mo>,</mo> <mi mathvariant="bold">a</mi> <mo>,</mo> <mi mathvariant="bold">b</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>z</mi> <mo>,</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, basis functions (<a href="#FD30-computation-11-00168" class="html-disp-formula">30</a>), and <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mrow> <mi mathvariant="bold">c</mi> <mo>,</mo> <mi mathvariant="bold">a</mi> <mo>,</mo> <mi mathvariant="bold">b</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>z</mi> <mo>,</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, basis functions (<a href="#FD31-computation-11-00168" class="html-disp-formula">31</a>), of exact solution <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>δ</mi> <mo>)</mo> </mrow> </semantics></math> to problems (<a href="#FD9-computation-11-00168" class="html-disp-formula">9</a>) and (<a href="#FD10-computation-11-00168" class="html-disp-formula">10</a>) with <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> neurons at <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (<b>left</b>) and <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>16</mn> </mrow> </semantics></math> (<b>right</b>).</p> Full article ">Figure 8
<p>Parameter dependence <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>i</mi> <msub> <mi>t</mi> <mrow> <mi>a</mi> <mi>d</mi> <mi>a</mi> <mi>p</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mn>0.1</mn> <mo>,</mo> <mn>1.5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD12-computation-11-00168" class="html-disp-formula">12</a>) and log(MSE) (<a href="#FD36-computation-11-00168" class="html-disp-formula">36</a>) of corresponding real-time adapting PINN, basis functions (<a href="#FD26-computation-11-00168" class="html-disp-formula">26</a>), and PINNp, basis functions (<a href="#FD27-computation-11-00168" class="html-disp-formula">27</a>), with 2 neurons and PPINN, basis functions (<a href="#FD30-computation-11-00168" class="html-disp-formula">30</a>), and PPINNp, basis functions (<a href="#FD31-computation-11-00168" class="html-disp-formula">31</a>), with 30 neurons of a hidden layer at each adaptation step <span class="html-italic">i</span> for tasks (<a href="#FD9-computation-11-00168" class="html-disp-formula">9</a>) and (<a href="#FD10-computation-11-00168" class="html-disp-formula">10</a>).</p> Full article ">Figure 9
<p>Parameter dependence <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>i</mi> <msub> <mi>t</mi> <mrow> <mi>a</mi> <mi>d</mi> <mi>a</mi> <mi>p</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mn>0.1</mn> <mo>,</mo> <mn>15</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD12-computation-11-00168" class="html-disp-formula">12</a>) and log(MSE) (<a href="#FD36-computation-11-00168" class="html-disp-formula">36</a>) of corresponding real-time adapting PINN(1), basis functions (<a href="#FD26-computation-11-00168" class="html-disp-formula">26</a>), with 2 neurons; PPINN(1), basis functions (<a href="#FD30-computation-11-00168" class="html-disp-formula">30</a>), with 30 neurons of a hidden layer at each adaptation step <span class="html-italic">i</span>; PINN(50), basis functions (<a href="#FD26-computation-11-00168" class="html-disp-formula">26</a>), with 2 neurons; and PPINN(50), basis functions (<a href="#FD30-computation-11-00168" class="html-disp-formula">30</a>), with 30 neurons. The data generated using the model (<a href="#FD24-computation-11-00168" class="html-disp-formula">24</a>), with the parameter change law during 50 iterations of training, occurring between the moments of data acquisition for task (<a href="#FD9-computation-11-00168" class="html-disp-formula">9</a>) and (<a href="#FD10-computation-11-00168" class="html-disp-formula">10</a>).</p> Full article ">Figure 10
<p>log(MSE) (<a href="#FD36-computation-11-00168" class="html-disp-formula">36</a>) of corresponding real-time adapting PINNp(1), basis functions (<a href="#FD27-computation-11-00168" class="html-disp-formula">27</a>), with 2 neurons; PPINNp(1), basis functions (<a href="#FD31-computation-11-00168" class="html-disp-formula">31</a>), with 30 neurons of a hidden layer at each adaptation step <span class="html-italic">i</span>; PINNp(50), basis functions (<a href="#FD27-computation-11-00168" class="html-disp-formula">27</a>), with 2 neurons; and PPINNp(50), basis functions (<a href="#FD31-computation-11-00168" class="html-disp-formula">31</a>), with 30 neurons. The data generated using the model (<a href="#FD24-computation-11-00168" class="html-disp-formula">24</a>) with the parameter change law during 50 iterations of training, occurring between the moments of data acquisition for task (<a href="#FD9-computation-11-00168" class="html-disp-formula">9</a>) and (<a href="#FD10-computation-11-00168" class="html-disp-formula">10</a>) parameter dependence <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>i</mi> <msub> <mi>t</mi> <mrow> <mi>a</mi> <mi>d</mi> <mi>a</mi> <mi>p</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mn>0.1</mn> <mo>,</mo> <mn>15</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD12-computation-11-00168" class="html-disp-formula">12</a>).</p> Full article ">Figure 11
<p>Parameter dependence <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mn>0.004</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD13-computation-11-00168" class="html-disp-formula">13</a>) and log(MSE) (<a href="#FD36-computation-11-00168" class="html-disp-formula">36</a>) of corresponding real-time adapting PINN, basis functions (<a href="#FD26-computation-11-00168" class="html-disp-formula">26</a>), with 2 neurons and PPINN, basis functions (<a href="#FD30-computation-11-00168" class="html-disp-formula">30</a>), with 30 neurons of a hidden layer at each adaptation step <span class="html-italic">i</span> for tasks (<a href="#FD9-computation-11-00168" class="html-disp-formula">9</a>) and (<a href="#FD10-computation-11-00168" class="html-disp-formula">10</a>); 5 and 500 iterations of training are considered between the moments of data reception.</p> Full article ">Figure 12
<p>Parameter dependence <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mn>0.004</mn> <mo>,</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD13-computation-11-00168" class="html-disp-formula">13</a>) and log(MSE) (<a href="#FD36-computation-11-00168" class="html-disp-formula">36</a>) of according real-time adapting PINN, basis functions (<a href="#FD26-computation-11-00168" class="html-disp-formula">26</a>), with 2 neurons and PPINN, basis functions (<a href="#FD30-computation-11-00168" class="html-disp-formula">30</a>), with 30 neurons of a hidden layer at each adaptation step <span class="html-italic">i</span> for tasks (<a href="#FD9-computation-11-00168" class="html-disp-formula">9</a>) and (<a href="#FD10-computation-11-00168" class="html-disp-formula">10</a>); 5 and 50 iterations of training are considered between the moments of data reception.</p> Full article ">Figure 13
<p>log(MSE) (<a href="#FD36-computation-11-00168" class="html-disp-formula">36</a>) of corresponding real-time adapting PINNp, basis functions (<a href="#FD27-computation-11-00168" class="html-disp-formula">27</a>), with 2 neurons and PPINNp, basis functions (<a href="#FD31-computation-11-00168" class="html-disp-formula">31</a>), with 30 neurons of a hidden layer at each adaptation step <span class="html-italic">i</span> for tasks (<a href="#FD9-computation-11-00168" class="html-disp-formula">9</a>) and (<a href="#FD10-computation-11-00168" class="html-disp-formula">10</a>); 5 and 500 iterations of training are considered between the moments of data reception, for parameter dependencies (<a href="#FD13-computation-11-00168" class="html-disp-formula">13</a>) <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mn>0.004</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (<b>left</b>), <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mn>0.004</mn> <mo>,</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (<b>right</b>).</p> Full article ">Figure 14
<p>The exact solution <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> to a system (<a href="#FD15-computation-11-00168" class="html-disp-formula">15</a>) at <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>, its two-neuron PINN (<a href="#FD3-computation-11-00168" class="html-disp-formula">3</a>) approximation <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="bold">c</mi> <mo>,</mo> <mi mathvariant="bold">a</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD41-computation-11-00168" class="html-disp-formula">41</a>), and the absolute error.</p> Full article ">Figure 15
<p>The exact solution <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> to a system (<a href="#FD15-computation-11-00168" class="html-disp-formula">15</a>) and their 10-neuron PPINN (<a href="#FD5-computation-11-00168" class="html-disp-formula">5</a>) approximation <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="bold">c</mi> <mo>,</mo> <mi mathvariant="bold">a</mi> <mo>,</mo> <mi mathvariant="bold">b</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD43-computation-11-00168" class="html-disp-formula">43</a>) at <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> (<b>left</b>) and <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> (<b>right</b>).</p> Full article ">Figure 16
<p>Parameter dependence <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>i</mi> <msub> <mi>t</mi> <mrow> <mi>a</mi> <mi>d</mi> <mi>a</mi> <mi>p</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mn>0.0045</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0.6</mn> <mo>,</mo> <mn>0.2</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and Log[MSE] (<a href="#FD44-computation-11-00168" class="html-disp-formula">44</a>) of corresponding real-time adapting PINN with 2 neurons and PPINN with 10 and 30 neurons of a hidden layer at each adaptation step <span class="html-italic">i</span>.</p> Full article ">Figure 17
<p>Parameter dependence <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>i</mi> <msub> <mi>t</mi> <mrow> <mi>a</mi> <mi>d</mi> <mi>a</mi> <mi>p</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mn>0.04</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and Log[MSE] (<a href="#FD44-computation-11-00168" class="html-disp-formula">44</a>) of corresponding real-time adapting PINN with 2 neurons and PPINN with 10 and 30 neurons of a hidden layer at each adaptation step <span class="html-italic">i</span>.</p> Full article ">Figure 18
<p>Parameter dependence <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>4</mn> </msub> <mrow> <mo>(</mo> <mi>i</mi> <msub> <mi>t</mi> <mrow> <mi>a</mi> <mi>d</mi> <mi>a</mi> <mi>p</mi> <mi>t</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.87</mn> </mrow> </semantics></math> and Log[MSE] (<a href="#FD44-computation-11-00168" class="html-disp-formula">44</a>) of according real-time adapting PINN with 2 neurons and PPINN with 10 and 30 neurons of a hidden layer at each adaptation step <span class="html-italic">i</span>.</p> Full article ">Figure 19
<p>The exact solution <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>δ</mi> <mo>)</mo> </mrow> </semantics></math> to a system (<a href="#FD20-computation-11-00168" class="html-disp-formula">20</a>) and its four-neuron PINN (<a href="#FD3-computation-11-00168" class="html-disp-formula">3</a>) approximation <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="bold">c</mi> <mo>,</mo> <mi mathvariant="bold">a</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD49-computation-11-00168" class="html-disp-formula">49</a>) at <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> on the diagonal <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </semantics></math> of <math display="inline"><semantics> <mo>Ω</mo> </semantics></math> (<b>left</b>) and the boundary <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (<b>right</b>).</p> Full article ">Figure 20
<p>The exact solution <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>δ</mi> <mo>)</mo> </mrow> </semantics></math> to a system (<a href="#FD20-computation-11-00168" class="html-disp-formula">20</a>) and its 50-neuron PPINN (<a href="#FD5-computation-11-00168" class="html-disp-formula">5</a>) approximation <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="bold">c</mi> <mo>,</mo> <mi mathvariant="bold">a</mi> <mo>,</mo> <mi mathvariant="bold">b</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> on the diagonal <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </semantics></math> of <math display="inline"><semantics> <mo>Ω</mo> </semantics></math> (<b>top left</b>) and the boundary <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (<b>top right</b>), on the diagonal <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (<b>bottom left</b>) and at <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> (<b>bottom right</b>).</p> Full article ">Figure 21
<p>Parameter dependence <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>i</mi> <msub> <mi>t</mi> <mrow> <mi>a</mi> <mi>d</mi> <mi>a</mi> <mi>p</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mn>0.003</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD22-computation-11-00168" class="html-disp-formula">22</a>) and Log[MSE] (<a href="#FD50-computation-11-00168" class="html-disp-formula">50</a>) of corresponding real-time adapting PINN with 4 neurons and PPINN with 50 neurons of a hidden layer at each adaptation step <span class="html-italic">i</span>.</p> Full article ">Figure 22
<p>Parameter dependence <math display="inline"><semantics> <mrow> <msub> <mi>δ</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>i</mi> <msub> <mi>t</mi> <mrow> <mi>a</mi> <mi>d</mi> <mi>a</mi> <mi>p</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mn>0.03</mn> <mo>,</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD22-computation-11-00168" class="html-disp-formula">22</a>) and Log[MSE] (<a href="#FD50-computation-11-00168" class="html-disp-formula">50</a>) of corresponding real-time adapting PINN with 4 neurons and PPINN with 50 neurons of a hidden layer at each adaptation step <span class="html-italic">i</span>.</p> Full article ">Figure 23
<p>Parameter dependence <math display="inline"><semantics> <mrow> <msub> <mi>δ</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>i</mi> <msub> <mi>t</mi> <mrow> <mi>a</mi> <mi>d</mi> <mi>a</mi> <mi>p</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mn>0.01</mn> <mo>,</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD23-computation-11-00168" class="html-disp-formula">23</a>) and Log[MSE] (<a href="#FD50-computation-11-00168" class="html-disp-formula">50</a>) of corresponding real-time adapting PINN with 4 neurons and PPINN with 50 neurons of a hidden layer at each adaptation step <span class="html-italic">i</span>.</p> Full article ">Figure 24
<p>Parameter dependence <math display="inline"><semantics> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>i</mi> <msub> <mi>t</mi> <mrow> <mi>a</mi> <mi>d</mi> <mi>a</mi> <mi>p</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mn>0.1</mn> <mo>,</mo> <mn>60</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD23-computation-11-00168" class="html-disp-formula">23</a>) and Log[MSE] (<a href="#FD50-computation-11-00168" class="html-disp-formula">50</a>) of corresponding real-time adapting PINN with 4 neurons and PPINN with 50 neurons of a hidden layer at each adaptation step <span class="html-italic">i</span>.</p> Full article ">Figure 25
<p>The exact solution <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>δ</mi> <mo>)</mo> </mrow> </semantics></math> to a system (<a href="#FD20-computation-11-00168" class="html-disp-formula">20</a>) and its 4-neuron PINN (<a href="#FD3-computation-11-00168" class="html-disp-formula">3</a>) approximation <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="bold">c</mi> <mo>,</mo> <mi mathvariant="bold">a</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD49-computation-11-00168" class="html-disp-formula">49</a>) (<b>left</b>) and 50-neuron PPINN (<a href="#FD5-computation-11-00168" class="html-disp-formula">5</a>) approximation <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="bold">c</mi> <mo>,</mo> <mi mathvariant="bold">a</mi> <mo>,</mo> <mi mathvariant="bold">b</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>δ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (<b>right</b>) on the diagonal <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </semantics></math> of <math display="inline"><semantics> <mo>Ω</mo> </semantics></math>, after stabilising <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>≈</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>. Adaptation with a small number of measurements (<math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>).</p> Full article ">
16 pages, 1491 KiB  
Article
Evolutionary PINN Learning Algorithms Inspired by Approximation to Pareto Front for Solving Ill-Posed Problems
by Tatiana Lazovskaya, Dmitriy Tarkhov, Maria Chistyakova, Egor Razumov, Anna Sergeeva and Tatiana Shemyakina
Computation 2023, 11(8), 166; https://doi.org/10.3390/computation11080166 - 21 Aug 2023
Cited by 1 | Viewed by 1341
Abstract
The article presents the development of new physics-informed evolutionary neural network learning algorithms. These algorithms aim to address the challenges of ill-posed problems by constructing a population close to the Pareto front. The study focuses on comparing the algorithm’s capabilities based on three [...] Read more.
The article presents the development of new physics-informed evolutionary neural network learning algorithms. These algorithms aim to address the challenges of ill-posed problems by constructing a population close to the Pareto front. The study focuses on comparing the algorithm’s capabilities based on three quality criteria of solutions. To evaluate the algorithms’ performance, two benchmark problems have been used. The first involved solving the Laplace equation in square regions with discontinuous boundary conditions. The second problem considered the absence of boundary conditions but with the presence of measurements. Additionally, the study investigates the influence of hyperparameters on the final results. Comparisons have been made between the proposed algorithms and standard algorithms for constructing neural networks based on physics (commonly referred to as vanilla’s algorithms). The results demonstrate the advantage of the proposed algorithms in achieving better performance when solving incorrectly posed problems. Furthermore, the proposed algorithms have the ability to identify specific solutions with the desired smoothness. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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Figure 1

Figure 1
<p>Evolutionary interpretation of Pareto mutation. The diagram showcases the construction of an analogue of the Pareto front of solutions, as well as the selection of a new generation of solutions through evolutionary processes. The yellow circles represent locations where specific problem criteria are incorporated by individuals’ evaluation.The training of PINNs is considered as Mutation 1–Mutation <math display="inline"><semantics> <msub> <mi>P</mi> <mn>1</mn> </msub> </semantics></math>.</p> Full article ">Figure 2
<p>The evolutionary algorithm used for generating a PINN solution population. The yellow circles represent locations where specific problem criteria are incorporated by individuals’ evaluation. The ‘Crossover’ refers to the process of introducing additional neurons to the current generation of PINNs.</p> Full article ">Figure 3
<p>A scheme of ’Crossover’ procedure. <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <msub> <mi>P</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> are the leftmost and rightmost ends of the current Pareto-optimal set with <span class="html-italic">n</span> neurons; <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>q</mi> <mo>)</mo> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>P</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </semantics></math> are current population of PINNs with extrernal multiplier (weight) <math display="inline"><semantics> <msub> <mi>c</mi> <mi>j</mi> </msub> </semantics></math> of each neuron. <math display="inline"><semantics> <mrow> <msub> <mi>c</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> are optimal external weights of new neuron minimising corresponding error <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mi>D</mi> </msub> <mrow> <mo>(</mo> <msub> <mi mathvariant="bold">X</mi> <msub> <mi>N</mi> <mi>D</mi> </msub> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>δ</mi> <msub> <mi>L</mi> <mi>B</mi> </msub> <mrow> <mo>(</mo> <msub> <mi mathvariant="bold">X</mi> <msub> <mi>N</mi> <mi>B</mi> </msub> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>. The yellow circle represents incorporating specific problem criteria by individuals’ evaluation.</p> Full article ">Figure 4
<p>Scatter plot shows all obtained and selected evolutionary PINN solutions for a two-objective optimisation problem (<a href="#FD10-computation-11-00166" class="html-disp-formula">10</a>)+(<a href="#FD11-computation-11-00166" class="html-disp-formula">11</a>), depending on the predefined number of neurons <span class="html-italic">N</span> in the final network.</p> Full article ">Figure 5
<p>Scatter plot shows all obtained and selected evolutionary PINN solutions for a two-objective optimisation problem (<a href="#FD10-computation-11-00166" class="html-disp-formula">10</a>)+(<a href="#FD11-computation-11-00166" class="html-disp-formula">11</a>), depending on the certain algorithm variant (1_1, 1_2 and 1_3).</p> Full article ">Figure 6
<p>Boxplots of root-mean-square error of satisfying the evolutionary PINN solutions for problem (<a href="#FD10-computation-11-00166" class="html-disp-formula">10</a>)+(<a href="#FD11-computation-11-00166" class="html-disp-formula">11</a>) equal to obtained using the Fourier method solution for different algorithm variations (1_1, 1_2 and 1_3).</p> Full article ">Figure 7
<p>Scatter plot shows all obtained and selected evolutionary PINN solutions for a two-objective optimisation problem (<a href="#FD10-computation-11-00166" class="html-disp-formula">10</a>)+(<a href="#FD11-computation-11-00166" class="html-disp-formula">11</a>), depending on the root-mean-square error ErrorDif of satisfying the derivative equal to zero on the upper boundary.</p> Full article ">Figure 8
<p>Scatter plot shows all obtained evolutionary PINN solutions for a two-objective optimisation problem (<a href="#FD10-computation-11-00166" class="html-disp-formula">10</a>)+(<a href="#FD12-computation-11-00166" class="html-disp-formula">12</a>), depending on the maximum of random measurement error.</p> Full article ">Figure 9
<p>Scatter plot shows all obtained evolutionary PINN solutions for a two-objective optimisation problem (<a href="#FD10-computation-11-00166" class="html-disp-formula">10</a>)+(<a href="#FD12-computation-11-00166" class="html-disp-formula">12</a>), depending on a common number of measurements.</p> Full article ">
28 pages, 548 KiB  
Article
The Complexity of the Super Subdivision of Cycle-Related Graphs Using Block Matrices
by Mohamed R. Zeen El Deen, Walaa A. Aboamer and Hamed M. El-Sherbiny
Computation 2023, 11(8), 162; https://doi.org/10.3390/computation11080162 - 15 Aug 2023
Cited by 1 | Viewed by 1231
Abstract
The complexity (number of spanning trees) in a finite graph Γ (network) is crucial. The quantity of spanning trees is a fundamental indicator for assessing the dependability of a network. The best and most dependable network is the one with the most spanning [...] Read more.
The complexity (number of spanning trees) in a finite graph Γ (network) is crucial. The quantity of spanning trees is a fundamental indicator for assessing the dependability of a network. The best and most dependable network is the one with the most spanning trees. In graph theory, one constantly strives to create novel structures from existing ones. The super subdivision operation produces more complicated networks, and the matrices of these networks can be divided into block matrices. Using methods from linear algebra and the characteristics of block matrices, we derive explicit formulas for determining the complexity of the super subdivision of a certain family of graphs, including the cycle Cn, where n=3,4,5,6; the dumbbell graph Dbm,n; the dragon graph Pm(Cn); the prism graph Πn, where n=3,4; the cycle Cn with a Pn2-chord, where n=4,6; and the complete graph K4. Additionally, 3D plots that were created using our results serve as illustrations. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Super subdivision <inline-formula><mml:math id="mm560"><mml:semantics><mml:mrow><mml:mi>S</mml:mi><mml:mi>S</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> of the cycle <inline-formula><mml:math id="mm561"><mml:semantics><mml:msub><mml:mi>C</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 2
<p>Super subdivision <inline-formula><mml:math id="mm562"><mml:semantics><mml:mrow><mml:mspace width="3.33333pt"/><mml:mi>S</mml:mi><mml:mi>S</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>D</mml:mi><mml:msub><mml:mi>b</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> of the dumbbell graph <inline-formula><mml:math id="mm563"><mml:semantics><mml:mrow><mml:mi>D</mml:mi><mml:msub><mml:mi>b</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 3
<p>Super subdivision <inline-formula><mml:math id="mm564"><mml:semantics><mml:mrow><mml:mi>S</mml:mi><mml:mi>S</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:msub><mml:mi>P</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> of the dragon graph <inline-formula><mml:math id="mm565"><mml:semantics><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 4
<p>Super subdivision <inline-formula><mml:math id="mm566"><mml:semantics><mml:mrow><mml:mi>S</mml:mi><mml:mi>S</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Π</mml:mo><mml:mn>3</mml:mn></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> of the prism <inline-formula><mml:math id="mm567"><mml:semantics><mml:msub><mml:mo>Π</mml:mo><mml:mn>3</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 5
<p>Super subdivision <inline-formula><mml:math id="mm568"><mml:semantics><mml:mrow><mml:mspace width="3.33333pt"/><mml:mi>S</mml:mi><mml:mi>S</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>⋇</mml:mo><mml:msub><mml:mi>P</mml:mi><mml:mfrac><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:mfrac></mml:msub><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> of the cycle <inline-formula><mml:math id="mm569"><mml:semantics><mml:msub><mml:mi>C</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> with a <inline-formula><mml:math id="mm570"><mml:semantics><mml:msub><mml:mi>P</mml:mi><mml:mfrac><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:mfrac></mml:msub></mml:semantics></mml:math></inline-formula> chord.</p> Full article ">Figure 6
<p>Super subdivision <inline-formula><mml:math id="mm571"><mml:semantics><mml:mrow><mml:mi>S</mml:mi><mml:mi>S</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> of the complete graph <inline-formula><mml:math id="mm572"><mml:semantics><mml:msub><mml:mi>K</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 7
<p>Variations in the enumerated complexities of the super subdivision graphs.</p> Full article ">
13 pages, 4351 KiB  
Article
Study on Optical Positioning Using Experimental Visible Light Communication System
by Nikoleta Vitsi, Argyris N. Stassinakis, Nikolaos A. Androutsos, George D. Roumelas, George K. Varotsos, Konstantinos Aidinis and Hector E. Nistazakis
Computation 2023, 11(8), 161; https://doi.org/10.3390/computation11080161 - 14 Aug 2023
Viewed by 1221
Abstract
Visible light positioning systems (VLP) have attracted significant commercial and research interest because of the many advantages they possess over other applications such as radio frequency (RF) positioning systems. In this work, an experimental configuration of an indoor VLP system based on the [...] Read more.
Visible light positioning systems (VLP) have attracted significant commercial and research interest because of the many advantages they possess over other applications such as radio frequency (RF) positioning systems. In this work, an experimental configuration of an indoor VLP system based on the well-known Lambertian light emission, is investigated. The corresponding results are also presented, and show that the system retains high enough accuracy to be operational, even in cases of low transmitted power and high background noise. Full article
(This article belongs to the Special Issue 10th Anniversary of Computation—Computational Engineering)
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Figure 1

Figure 1
<p>Optical transmitter and receiver. All the parameter values appear in <a href="#computation-11-00161-t001" class="html-table">Table 1</a>.</p> Full article ">Figure 2
<p>Experimental setup schematic.</p> Full article ">Figure 3
<p>Positioning experimental setup.</p> Full article ">Figure 4
<p>Pulses of positioning system.</p> Full article ">Figure 5
<p>Regression fitting results.</p> Full article ">Figure 6
<p>Signal received in each case.</p> Full article ">Figure 7
<p>Case 1 results.</p> Full article ">Figure 8
<p>Case 2 results.</p> Full article ">Figure 9
<p>Case 3 results.</p> Full article ">Figure 10
<p>Case 4 results.</p> Full article ">
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