Mathematical and Computational Applications

15 pages, 5025 KiB

Open AccessArticle

Analytical Equations Applied to the Study of Steel Profiles under Fire According to Different Nominal Temperature-Time Curves

by Pedro N. Oliveira, Elza M. M. Fonseca, Raul D. S. G. Campilho and Paulo A. G. Piloto

Math. Comput. Appl. 2021, 26(2), 48; https://doi.org/10.3390/mca26020048 - 18 Jun 2021

Cited by 8 | Viewed by 3454

Abstract

Some analytical methods are available for temperature evaluation in solid bodies. These methods can be used due to their simplicity and good results. The main goal of this work is to present the temperature calculation in different cross-sections of structural hot-rolled steel profiles [...] Read more.

Some analytical methods are available for temperature evaluation in solid bodies. These methods can be used due to their simplicity and good results. The main goal of this work is to present the temperature calculation in different cross-sections of structural hot-rolled steel profiles (IPE, HEM, L, and UAP) using the lumped capacitance method and the simplified equation from Eurocode 3. The basis of the lumped capacitance method is that the temperature of the solid body is uniform at any given time instant during a heat transient process. The profiles were studied, subjected to the fire action according to the nominal temperature–time curves (standard temperature-time curve ISO 834, external fire curve, and hydrocarbon fire curve). The obtained results allow verifying the agreement between the two methodologies and the influence in the temperature field due to the use of different nominal fire curves. This finding enables us to conclude that the lumped capacitance method is accurate and could be easily applied. Full article

(This article belongs to the Special Issue Numerical and Symbolic Computation: Developments and Applications 2021)

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37 pages, 16808 KiB

Open AccessFeature PaperEditor’s ChoiceReview

Review of Multi-Physics Modeling on the Active Magnetic Regenerative Refrigeration

by Julien Eustache, Antony Plait, Frédéric Dubas and Raynal Glises

Math. Comput. Appl. 2021, 26(2), 47; https://doi.org/10.3390/mca26020047 - 15 Jun 2021

Cited by 6 | Viewed by 3642

Abstract

Compared to conventional vapor-compression refrigeration systems, magnetic refrigeration is a promising and potential alternative technology. The magnetocaloric effect (MCE) is used to produce heat and cold sources through a magnetocaloric material (MCM). The material is submitted to a magnetic field with active magnetic [...] Read more.

Compared to conventional vapor-compression refrigeration systems, magnetic refrigeration is a promising and potential alternative technology. The magnetocaloric effect (MCE) is used to produce heat and cold sources through a magnetocaloric material (MCM). The material is submitted to a magnetic field with active magnetic regenerative refrigeration (AMRR) cycles. Initially, this effect was widely used for cryogenic applications to achieve very low temperatures. However, this technology must be improved to replace vapor-compression devices operating around room temperature. Therefore, over the last 30 years, a lot of studies have been done to obtain more efficient devices. Thus, the modeling is a crucial step to perform a preliminary study and optimization. In this paper, after a large introduction on MCE research, a state-of-the-art of multi-physics modeling on the AMRR cycle modeling is made. To end this paper, a suggestion of innovative and advanced modeling solutions to study magnetocaloric regenerator is described. Full article

(This article belongs to the Special Issue Characterization of Magnetocaloric Devices and Materials through Mathematical Models)

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16 pages, 16590 KiB

Open AccessArticle

Optimization of Power Generation Grids: A Case of Study in Eastern Mexico

by Esmeralda López, René F. Domínguez-Cruz and Iván Salgado-Tránsito

Math. Comput. Appl. 2021, 26(2), 46; https://doi.org/10.3390/mca26020046 - 8 Jun 2021

Cited by 2 | Viewed by 2843

Abstract

Optimization of energy resources is a priority issue for our society. An improper imbalance between demand and power generation can lead to inefficient use of installed capacity, waste of fuels, worse effects on the environment, and higher costs. This paper presents the preliminary [...] Read more.

Optimization of energy resources is a priority issue for our society. An improper imbalance between demand and power generation can lead to inefficient use of installed capacity, waste of fuels, worse effects on the environment, and higher costs. This paper presents the preliminary results of a study of seventeen interconnected power generation plants situated in eastern Mexico. The aim of the research is to apply a linear programming model to find the system-optimal solution by minimizing operating costs for this grid of power plants. The calculations were made taking into account the actual parameters of each plant; the demand and production of energy were analyzed in four time periods of 6 h during a day. The results show the cost-optimal configuration of the current power infrastructure obtained from a simple implementation model in MATLAB^® software. The contribution of this paper is to adapt a lineal progamming model for an electrical distribution network formed with different types of power generation technology. The study shows that fossil fuel plants, besides emitting greenhouse gases that affect human health and the environment, incur maintenance expenses even without operation. The results are a helpful instrument for decision-making regarding the rational use of available installed capacity. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)

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28 pages, 2919 KiB

Open AccessArticle

Variational Bayesian Learning of SMoGs: Modelling and Their Application to Synthetic Aperture Radar

by Evangelos Roussos

Math. Comput. Appl. 2021, 26(2), 45; https://doi.org/10.3390/mca26020045 - 7 Jun 2021

Viewed by 2298

Abstract

We show how modern Bayesian Machine Learning tools can be effectively used in order to develop efficient methods for filtering Earth Observation signals. Bayesian statistical methods can be thought of as a generalization of the classical least-squares adjustment methods where both the unknown [...] Read more.

We show how modern Bayesian Machine Learning tools can be effectively used in order to develop efficient methods for filtering Earth Observation signals. Bayesian statistical methods can be thought of as a generalization of the classical least-squares adjustment methods where both the unknown signals and the parameters are endowed with probability distributions, the priors. Statistical inference under this scheme is the derivation of posterior distributions, that is, distributions of the unknowns after the model has seen the data. Least squares can then be thought of as a special case that uses Gaussian likelihoods, or error statistics. In principle, for most non-trivial models, this framework requires performing integration in high-dimensional spaces. Variational methods are effective tools for approximate inference in Statistical Machine Learning and Computational Statistics. In this paper, after introducing the general variational Bayesian learning method, we apply it to the modelling and implementation of sparse mixtures of Gaussians (SMoG) models, intended to be used as adaptive priors for the efficient representation of sparse signals in applications such as wavelet-type analysis. Wavelet decomposition methods have been very successful in denoising real-world, non-stationary signals that may also contain discontinuities. For this purpose we construct a constrained hierarchical Bayesian model capturing the salient characteristics of such sets of decomposition coefficients. We express our model as a Dirichlet mixture model. We then show how variational ideas can be used to derive efficient methods for bypassing the need for integration: the task of integration becomes one of optimization. We apply our SMoG implementation to the problem of denoising of Synthetic Aperture Radar images, inherently affected by speckle noise, and show that it achieves improved performance compared to established methods, both in terms of speckle reduction and image feature preservation. Full article

(This article belongs to the Special Issue Mathematical Modelling in Engineering & Human Behaviour 2019)

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24 pages, 4238 KiB

Open AccessFeature PaperArticle

Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients

by Eric Chung, Hyea-Hyun Kim, Ming-Fai Lam and Lina Zhao

Math. Comput. Appl. 2021, 26(2), 44; https://doi.org/10.3390/mca26020044 - 6 Jun 2021

Cited by 1 | Viewed by 3045

Abstract

In this paper, we consider the balancing domain decomposition by constraints (BDDC) algorithm with adaptive coarse spaces for a class of stochastic elliptic problems. The key ingredient in the construction of the coarse space is the solutions of local spectral problems, which depend [...] Read more.

In this paper, we consider the balancing domain decomposition by constraints (BDDC) algorithm with adaptive coarse spaces for a class of stochastic elliptic problems. The key ingredient in the construction of the coarse space is the solutions of local spectral problems, which depend on the coefficient of the PDE. This poses a significant challenge for stochastic coefficients as it is computationally expensive to solve the local spectral problems for every realization of the coefficient. To tackle this computational burden, we propose a machine learning approach. Our method is based on the use of a deep neural network (DNN) to approximate the relation between the stochastic coefficients and the coarse spaces. For the input of the DNN, we apply the Karhunen–Loève expansion and use the first few dominant terms in the expansion. The output of the DNN is the resulting coarse space, which is then applied with the standard adaptive BDDC algorithm. We will present some numerical results with oscillatory and high contrast coefficients to show the efficiency and robustness of the proposed scheme. Full article

(This article belongs to the Special Issue Domain Decomposition Methods)

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19 pages, 3482 KiB

Open AccessFeature PaperArticle

Simple Algebraic Expressions for the Prediction and Control of High-Temperature Annealed Structures by Linear Perturbation Analysis

by Constantino Grau Turuelo and Cornelia Breitkopf

Math. Comput. Appl. 2021, 26(2), 43; https://doi.org/10.3390/mca26020043 - 1 Jun 2021

Cited by 2 | Viewed by 2535

Abstract

The prediction and control of the transformation of void structures with high-temperature processing is a critical area in many engineering applications. In this work, focused on the void shape evolution of silicon, a novel algebraic model for the calculation of final equilibrium structures [...] Read more.

The prediction and control of the transformation of void structures with high-temperature processing is a critical area in many engineering applications. In this work, focused on the void shape evolution of silicon, a novel algebraic model for the calculation of final equilibrium structures from initial void cylindrical trenches, driven by surface diffusion, is introduced. This algebraic model provides a simple and fast way to calculate expressions to predict the final geometrical characteristics, based on linear perturbation analysis. The obtained results are similar to most compared literature data, especially, to those in which a final transformation is reached. Additionally, the model can be applied in any materials affected by the surface diffusion. With such a model, the calculation of void structure design points is greatly simplified not only in the semiconductors field but in other engineering fields where surface diffusion phenomenon is studied. Full article

(This article belongs to the Special Issue Mathematical and Computational Modelling in Mechanics of Materials and Structures)

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13 pages, 527 KiB

Open AccessArticle

Effectiveness of Floating-Point Precision on the Numerical Approximation by Spectral Methods

by José A. O. Matos and Paulo B. Vasconcelos

Math. Comput. Appl. 2021, 26(2), 42; https://doi.org/10.3390/mca26020042 - 26 May 2021

Cited by 1 | Viewed by 3319

Abstract

With the fast advances in computational sciences, there is a need for more accurate computations, especially in large-scale solutions of differential problems and long-term simulations. Amid the many numerical approaches to solving differential problems, including both local and global methods, spectral methods can [...] Read more.

With the fast advances in computational sciences, there is a need for more accurate computations, especially in large-scale solutions of differential problems and long-term simulations. Amid the many numerical approaches to solving differential problems, including both local and global methods, spectral methods can offer greater accuracy. The downside is that spectral methods often require high-order polynomial approximations, which brings numerical instability issues to the problem resolution. In particular, large condition numbers associated with the large operational matrices, prevent stable algorithms from working within machine precision. Software-based solutions that implement arbitrary precision arithmetic are available and should be explored to obtain higher accuracy when needed, even with the higher computing time cost associated. In this work, experimental results on the computation of approximate solutions of differential problems via spectral methods are detailed with recourse to quadruple precision arithmetic. Variable precision arithmetic was used in Tau Toolbox, a mathematical software package to solve integro-differential problems via the spectral Tau method. Full article

(This article belongs to the Special Issue Numerical and Symbolic Computation: Developments and Applications 2021)

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24 pages, 3666 KiB

Open AccessArticle

Semi-Analytical Solution of Two-Dimensional Viscous Flow through Expanding/Contracting Gaps with Permeable Walls

by Mohammad Mehdi Rashidi, Mikhail A. Sheremet, Maryam Sadri, Satyaranjan Mishra, Pradyumna Kumar Pattnaik, Faranak Rabiei, Saeid Abbasbandy, Hussein Sahihi and Esmaeel Erfani

Math. Comput. Appl. 2021, 26(2), 41; https://doi.org/10.3390/mca26020041 - 23 May 2021

Cited by 12 | Viewed by 2870

Abstract

In this research, the analytical methods of the differential transform method (DTM), homotopy asymptotic method (HAM), optimal homotopy asymptotic method (OHAM), Adomian decomposition method (ADM), variation iteration method (VIM) and reproducing kernel Hilbert space method (RKHSM), and the numerical method of the finite [...] Read more.

In this research, the analytical methods of the differential transform method (DTM), homotopy asymptotic method (HAM), optimal homotopy asymptotic method (OHAM), Adomian decomposition method (ADM), variation iteration method (VIM) and reproducing kernel Hilbert space method (RKHSM), and the numerical method of the finite difference method (FDM) for (analytical-numerical) simulation of 2D viscous flow along expanding/contracting channels with permeable borders are carried out. The solutions for analytical method are obtained in series form (and the series are convergent), while for the numerical method the solution is obtained taking into account approximation techniques of second-order accuracy. The OHAM and HAM provide an appropriate method for controlling the convergence of the discretization series and adjusting convergence domains, despite having a problem for large sizes of obtained results in series form; for instance, the size of the series solution for the DTM is very small for the same order of accuracy. It is hard to judge which method is the best and all of them have their advantages and disadvantages. For instance, applying the DTM to BVPs is difficult; however, solving BVPs with the HAM, OHAM and VIM is simple and straightforward. The extracted solutions, in comparison with the computational solutions (shooting procedure combined with a Runge–Kutta fourth-order scheme, finite difference method), demonstrate remarkable accuracy. Finally, CPU time, average error and residual error for different cases are presented in tables and figures. Full article

(This article belongs to the Special Issue Advances in Computational Fluid Dynamics and Heat & Mass Transfer)

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20 pages, 5615 KiB

Open AccessFeature PaperArticle

Semi-Supervised Learning Using Hierarchical Mixture Models: Gene Essentiality Case Study

by Michael W. Daniels, Daniel Dvorkin, Rani K. Powers and Katerina Kechris

Math. Comput. Appl. 2021, 26(2), 40; https://doi.org/10.3390/mca26020040 - 18 May 2021

Cited by 1 | Viewed by 2493

Abstract

Integrating gene-level data is useful for predicting the role of genes in biological processes. This problem has typically focused on supervised classification, which requires large training sets of positive and negative examples. However, training data sets that are too small for supervised approaches [...] Read more.

Integrating gene-level data is useful for predicting the role of genes in biological processes. This problem has typically focused on supervised classification, which requires large training sets of positive and negative examples. However, training data sets that are too small for supervised approaches can still provide valuable information. We describe a hierarchical mixture model that uses limited positively labeled gene training data for semi-supervised learning. We focus on the problem of predicting essential genes, where a gene is required for the survival of an organism under particular conditions. We applied cross-validation and found that the inclusion of positively labeled samples in a semi-supervised learning framework with the hierarchical mixture model improves the detection of essential genes compared to unsupervised, supervised, and other semi-supervised approaches. There was also improved prediction performance when genes are incorrectly assumed to be non-essential. Our comparisons indicate that the incorporation of even small amounts of existing knowledge improves the accuracy of prediction and decreases variability in predictions. Although we focused on gene essentiality, the hierarchical mixture model and semi-supervised framework is standard for problems focused on prediction of genes or other features, with multiple data types characterizing the feature, and a small set of positive labels. Full article

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21 pages, 9831 KiB

Open AccessArticle

A Peptides Prediction Methodology for Tertiary Structure Based on Simulated Annealing

by Juan P. Sánchez-Hernández, Juan Frausto-Solís, Juan J. González-Barbosa, Diego A. Soto-Monterrubio, Fanny G. Maldonado-Nava and Guadalupe Castilla-Valdez

Math. Comput. Appl. 2021, 26(2), 39; https://doi.org/10.3390/mca26020039 - 29 Apr 2021

Cited by 3 | Viewed by 3567

Abstract

The Protein Folding Problem (PFP) is a big challenge that has remained unsolved for more than fifty years. This problem consists of obtaining the tertiary structure or Native Structure (NS) of a protein knowing its amino acid sequence. The computational methodologies applied to [...] Read more.

The Protein Folding Problem (PFP) is a big challenge that has remained unsolved for more than fifty years. This problem consists of obtaining the tertiary structure or Native Structure (NS) of a protein knowing its amino acid sequence. The computational methodologies applied to this problem are classified into two groups, known as Template-Based Modeling (TBM) and ab initio models. In the latter methodology, only information from the primary structure of the target protein is used. In the literature, Hybrid Simulated Annealing (HSA) algorithms are among the best ab initio algorithms for PFP; Golden Ratio Simulated Annealing (GRSA) is a PFP family of these algorithms designed for peptides. Moreover, for the algorithms designed with TBM, they use information from a target protein’s primary structure and information from similar or analog proteins. This paper presents GRSA-SSP methodology that implements a secondary structure prediction to build an initial model and refine it with HSA algorithms. Additionally, we compare the performance of the GRSAX-SSP algorithms versus its corresponding GRSAX. Finally, our best algorithm GRSAX-SSP is compared with PEP-FOLD3, I-TASSER, QUARK, and Rosetta, showing that it competes in small peptides except when predicting the largest peptides. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)

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24 pages, 2956 KiB

Open AccessArticle

Operational Risk Reverse Stress Testing: Optimal Solutions

by Peter Mitic

Math. Comput. Appl. 2021, 26(2), 38; https://doi.org/10.3390/mca26020038 - 28 Apr 2021

Cited by 2 | Viewed by 3023

Abstract

Selecting a suitable method to solve a black-box optimization problem that uses noisy data was considered. A targeted stop condition for the function to be optimized, implemented as a stochastic algorithm, makes established Bayesian methods inadmissible. A simple modification was proposed and shown [...] Read more.

Selecting a suitable method to solve a black-box optimization problem that uses noisy data was considered. A targeted stop condition for the function to be optimized, implemented as a stochastic algorithm, makes established Bayesian methods inadmissible. A simple modification was proposed and shown to improve optimization the efficiency considerably. The optimization effectiveness was measured in terms of the mean and standard deviation of the number of function evaluations required to achieve the target. Comparisons with alternative methods showed that the modified Bayesian method and binary search were both performant, but in different ways. In a sequence of identical runs, the former had a lower expected value for the number of runs needed to find an optimal value. The latter had a lower standard deviation for the same sequence of runs. Additionally, we suggested a way to find an approximate solution to the same problem using symbolic computation. Faster results could be obtained at the expense of some impaired accuracy and increased memory requirements. Full article

(This article belongs to the Special Issue Numerical and Symbolic Computation: Developments and Applications 2021)

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31 pages, 673 KiB

Open AccessFeature PaperArticle

An Evolutionary View of the U.S. Supreme Court

by Noah Giansiracusa

Math. Comput. Appl. 2021, 26(2), 37; https://doi.org/10.3390/mca26020037 - 28 Apr 2021

Cited by 1 | Viewed by 2781

Abstract

The voting patterns of the nine justices on the United States Supreme Court continue to fascinate and perplex observers of the Court. While it is commonly understood that the division of the justices into a liberal branch and a conservative branch inevitably drives [...] Read more.

The voting patterns of the nine justices on the United States Supreme Court continue to fascinate and perplex observers of the Court. While it is commonly understood that the division of the justices into a liberal branch and a conservative branch inevitably drives many case outcomes, there are finer, less transparent divisions within these two main branches that have proven difficult to extract empirically. This study imports methods from evolutionary biology to help illuminate the intricate and often overlooked branching structure of the justices’ voting behavior. Specifically, phylogenetic tree estimation based on voting disagreement rates is used to extend ideal point estimation to the non-Euclidean setting of hyperbolic metrics. After introducing this framework, comparing it to one- and two-dimensional multidimensional scaling, and arguing that it flexibly captures important higher-dimensional voting behavior, a handful of potential ways to apply this tool are presented. The emphasis throughout is on interpreting these judicial trees and extracting qualitative insights from them. Full article

(This article belongs to the Section Social Sciences)

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30 pages, 2237 KiB

Open AccessArticle

Modeling and Optimizing the Multi-Objective Portfolio Optimization Problem with Trapezoidal Fuzzy Parameters

by Alejandro Estrada-Padilla, Daniela Lopez-Garcia, Claudia Gómez-Santillán, Héctor Joaquín Fraire-Huacuja, Laura Cruz-Reyes, Nelson Rangel-Valdez and María Lucila Morales-Rodríguez

Math. Comput. Appl. 2021, 26(2), 36; https://doi.org/10.3390/mca26020036 - 24 Apr 2021

Cited by 4 | Viewed by 2857

Abstract

A common issue in the Multi-Objective Portfolio Optimization Problem (MOPOP) is the presence of uncertainty that affects individual decisions, e.g., variations on resources or benefits of projects. Fuzzy numbers are successful in dealing with imprecise numerical quantities, and they found numerous applications in [...] Read more.

A common issue in the Multi-Objective Portfolio Optimization Problem (MOPOP) is the presence of uncertainty that affects individual decisions, e.g., variations on resources or benefits of projects. Fuzzy numbers are successful in dealing with imprecise numerical quantities, and they found numerous applications in optimization. However, so far, they have not been used to tackle uncertainty in MOPOP. Hence, this work proposes to tackle MOPOP’s uncertainty with a new optimization model based on fuzzy trapezoidal parameters. Additionally, it proposes three novel steady-state algorithms as the model’s solution process. One approach integrates the Fuzzy Adaptive Multi-objective Evolutionary (FAME) methodology; the other two apply the Non-Dominated Genetic Algorithm (NSGA-II) methodology. One steady-state algorithm uses the Spatial Spread Deviation as a density estimator to improve the Pareto fronts’ distribution. This research work’s final contribution is developing a new defuzzification mapping that allows measuring algorithms’ performance using widely known metrics. The results show a significant difference in performance favoring the proposed steady-state algorithm based on the FAME methodology. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)

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35 pages, 851 KiB

Open AccessArticle

An Interactive Recommendation System for Decision Making Based on the Characterization of Cognitive Tasks

by Teodoro Macias-Escobar, Laura Cruz-Reyes, César Medina-Trejo, Claudia Gómez-Santillán, Nelson Rangel-Valdez and Héctor Fraire-Huacuja

Math. Comput. Appl. 2021, 26(2), 35; https://doi.org/10.3390/mca26020035 - 21 Apr 2021

Cited by 5 | Viewed by 3044

Abstract

The decision-making process can be complex and underestimated, where mismanagement could lead to poor results and excessive spending. This situation appears in highly complex multi-criteria problems such as the project portfolio selection (PPS) problem. Therefore, a recommender system becomes crucial to guide the [...] Read more.

The decision-making process can be complex and underestimated, where mismanagement could lead to poor results and excessive spending. This situation appears in highly complex multi-criteria problems such as the project portfolio selection (PPS) problem. Therefore, a recommender system becomes crucial to guide the solution search process. To our knowledge, most recommender systems that use argumentation theory are not proposed for multi-criteria optimization problems. Besides, most of the current recommender systems focused on PPS problems do not attempt to justify their recommendations. This work studies the characterization of cognitive tasks involved in the decision-aiding process to propose a framework for the Decision Aid Interactive Recommender System (DAIRS). The proposed system focuses on a user-system interaction that guides the search towards the best solution considering a decision-maker’s preferences. The developed framework uses argumentation theory supported by argumentation schemes, dialogue games, proof standards, and two state transition diagrams (STD) to generate and explain its recommendations to the user. This work presents a prototype of DAIRS to evaluate the user experience on multiple real-life case simulations through a usability measurement. The prototype and both STDs received a satisfying score and mostly overall acceptance by the test users. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)

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18 pages, 1780 KiB

Open AccessFeature PaperArticle

A Sequential Approach for Aerodynamic Shape Optimization with Topology Optimization of Airfoils

by Isaac Gibert Martínez, Frederico Afonso, Simão Rodrigues and Fernando Lau

Math. Comput. Appl. 2021, 26(2), 34; https://doi.org/10.3390/mca26020034 - 20 Apr 2021

Cited by 3 | Viewed by 4690

Abstract

The objective of this work is to study the coupling of two efficient optimization techniques, Aerodynamic Shape Optimization (ASO) and Topology Optimization (TO), in 2D airfoils. To achieve such goal two open-source codes, SU2 and Calculix, are employed for ASO and TO, respectively, [...] Read more.

The objective of this work is to study the coupling of two efficient optimization techniques, Aerodynamic Shape Optimization (ASO) and Topology Optimization (TO), in 2D airfoils. To achieve such goal two open-source codes, SU2 and Calculix, are employed for ASO and TO, respectively, using the Sequential Least SQuares Programming (SLSQP) and the Bi-directional Evolutionary Structural Optimization (BESO) algorithms; the latter is well-known for allowing the addition of material in the TO which constitutes, as far as our knowledge, a novelty for this kind of application. These codes are linked by means of a script capable of reading the geometry and pressure distribution obtained from the ASO and defining the boundary conditions to be applied in the TO. The Free-Form Deformation technique is chosen for the definition of the design variables to be used in the ASO, while the densities of the inner elements are defined as design variables of the TO. As a test case, a widely used benchmark transonic airfoil, the RAE2822, is chosen here with an internal geometric constraint to simulate the wing-box of a transonic wing. First, the two optimization procedures are tested separately to gain insight and then are run in a sequential way for two test cases with available experimental data: (i) Mach 0.729 at

α = {2.31}^{°}

; and (ii) Mach 0.730 at

α = {2.79}^{°}

. In the ASO problem, the lift is fixed and the drag is minimized; while in the TO problem, compliance minimization is set as the objective for a prescribed volume fraction. Improvements in both aerodynamic and structural performance are found, as expected: the ASO reduced the total pressure on the airfoil surface in order to minimize drag, which resulted in lower stress values experienced by the structure. Full article

(This article belongs to the Special Issue Numerical and Symbolic Computation: Developments and Applications 2021)

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12 pages, 1000 KiB

Open AccessArticle

Nonlinear and Stability Analysis of a Ship with General Roll-Damping Using an Asymptotic Perturbation Method

by Muhammad Usman, Shaaban Abdallah and Mudassar Imran

Math. Comput. Appl. 2021, 26(2), 33; https://doi.org/10.3390/mca26020033 - 20 Apr 2021

Viewed by 2573

Abstract

In this work, the response of a ship rolling in regular beam waves is studied. The model is one degree of freedom model for nonlinear ship dynamics. The model consists of the terms containing inertia, damping, restoring forces, and external forces. The asymptotic [...] Read more.

In this work, the response of a ship rolling in regular beam waves is studied. The model is one degree of freedom model for nonlinear ship dynamics. The model consists of the terms containing inertia, damping, restoring forces, and external forces. The asymptotic perturbation method is used to study the primary resonance phenomena. The effects of various parameters are studied on the stability of steady states. It is shown that the variation of bifurcation parameters affects the bending of the bifurcation curve. The slope stability theorems are also presented. Full article

(This article belongs to the Special Issue Mathematical and Computational Modelling in Mechanics of Materials and Structures)

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23 pages, 746 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

ROM-Based Inexact Subdivision Methods for PDE-Constrained Multiobjective Optimization

by Stefan Banholzer, Bennet Gebken, Lena Reichle and Stefan Volkwein

Math. Comput. Appl. 2021, 26(2), 32; https://doi.org/10.3390/mca26020032 - 15 Apr 2021

Cited by 1 | Viewed by 2362

Abstract

The goal in multiobjective optimization is to determine the so-called Pareto set. Our optimization problem is governed by a parameter-dependent semi-linear elliptic partial differential equation (PDE). To solve it, we use a gradient-based set-oriented numerical method. The numerical solution of the PDE by [...] Read more.

The goal in multiobjective optimization is to determine the so-called Pareto set. Our optimization problem is governed by a parameter-dependent semi-linear elliptic partial differential equation (PDE). To solve it, we use a gradient-based set-oriented numerical method. The numerical solution of the PDE by standard discretization methods usually leads to high computational effort. To overcome this difficulty, reduced-order modeling (ROM) is developed utilizing the reduced basis method. These model simplifications cause inexactness in the gradients. For that reason, an additional descent condition is proposed. Applying a modified subdivision algorithm, numerical experiments illustrate the efficiency of our solution approach. Full article

(This article belongs to the Special Issue Set Oriented Numerics 2022)

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37 pages, 1551 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Derivative-Free Multiobjective Trust Region Descent Method Using Radial Basis Function Surrogate Models

by Manuel Berkemeier and Sebastian Peitz

Math. Comput. Appl. 2021, 26(2), 31; https://doi.org/10.3390/mca26020031 - 15 Apr 2021

Cited by 5 | Viewed by 3398

Abstract

We present a local trust region descent algorithm for unconstrained and convexly constrained multiobjective optimization problems. It is targeted at heterogeneous and expensive problems, i.e., problems that have at least one objective function that is computationally expensive. Convergence to a Pareto critical point [...] Read more.

We present a local trust region descent algorithm for unconstrained and convexly constrained multiobjective optimization problems. It is targeted at heterogeneous and expensive problems, i.e., problems that have at least one objective function that is computationally expensive. Convergence to a Pareto critical point is proven. The method is derivative-free in the sense that derivative information need not be available for the expensive objectives. Instead, a multiobjective trust region approach is used that works similarly to its well-known scalar counterparts and complements multiobjective line-search algorithms. Local surrogate models constructed from evaluation data of the true objective functions are employed to compute possible descent directions. In contrast to existing multiobjective trust region algorithms, these surrogates are not polynomial but carefully constructed radial basis function networks. This has the important advantage that the number of data points needed per iteration scales linearly with the decision space dimension. The local models qualify as fully linear and the corresponding general scalar framework is adapted for problems with multiple objectives. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)

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16 pages, 1422 KiB

Open AccessArticle

A Front-Fixing Implicit Finite Difference Method for the American Put Options Model

by Riccardo Fazio, Alessandra Insana and Alessandra Jannelli

Math. Comput. Appl. 2021, 26(2), 30; https://doi.org/10.3390/mca26020030 - 13 Apr 2021

Cited by 2 | Viewed by 2917

Abstract

In this paper, we present an implicit finite difference method for the numerical solution of the Black–Scholes model of American put options without dividend payments. We combine the proposed numerical method by using a front-fixing approach where the option price and the early [...] Read more.

In this paper, we present an implicit finite difference method for the numerical solution of the Black–Scholes model of American put options without dividend payments. We combine the proposed numerical method by using a front-fixing approach where the option price and the early exercise boundary are computed simultaneously. We study the consistency and prove the stability of the implicit method by fixing the values of the free boundary and of its first derivative. We improve the accuracy of the computed solution via a mesh refinement based on Richardson’s extrapolation. Comparisons with some proposed methods for the American options problem are carried out to validate the obtained numerical results and to show the efficiency of the proposed numerical method. Finally, by using an a posteriori error estimator, we find a suitable computational grid requiring that the computed solution verifies a prefixed error tolerance. Full article

(This article belongs to the Special Issue Mathematical and Computational Applications in Finance and Economics)

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11 pages, 573 KiB

Open AccessArticle

Convolutional Neural Network–Component Transformation (CNN–CT) for Confirmed COVID-19 Cases

by Juan Frausto-Solís, Lucía J. Hernández-González, Juan J. González-Barbosa, Juan Paulo Sánchez-Hernández and Edgar Román-Rangel

Math. Comput. Appl. 2021, 26(2), 29; https://doi.org/10.3390/mca26020029 - 12 Apr 2021

Cited by 8 | Viewed by 3181

Abstract

The COVID-19 disease constitutes a global health contingency. This disease has left millions people infected, and its spread has dramatically increased. This study proposes a new method based on a Convolutional Neural Network (CNN) and temporal Component Transformation (CT) called CNN–CT. This method [...] Read more.

The COVID-19 disease constitutes a global health contingency. This disease has left millions people infected, and its spread has dramatically increased. This study proposes a new method based on a Convolutional Neural Network (CNN) and temporal Component Transformation (CT) called CNN–CT. This method is applied to confirmed cases of COVID-19 in the United States, Mexico, Brazil, and Colombia. The CT changes daily predictions and observations to weekly components and vice versa. In addition, CNN–CT adjusts the predictions made by CNN using AutoRegressive Integrated Moving Average (ARIMA) and Exponential Smoothing (ES) methods. This combination of strategies provides better predictions than most of the individual methods by themselves. In this paper, we present the mathematical formulation for this strategy. Our experiments encompass the fine-tuning of the parameters of the algorithms. We compared the best hybrid methods obtained with CNN–CT versus the individual CNN, Long Short-Term Memory (LSTM), ARIMA, and ES methods. Our results show that our hybrid method surpasses the performance of LSTM, and that it consistently achieves competitive results in terms of the MAPE metric, as opposed to the individual CNN and ARIMA methods, whose performance varies largely for different scenarios. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)

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21 pages, 4990 KiB

Open AccessArticle

Effect of the Profile of the Decision Maker in the Search for Solutions in the Decision-Making Process

by Mercedes Perez-Villafuerte, Laura Cruz-Reyes, Nelson Rangel-Valdez, Claudia Gomez-Santillan and Héctor Fraire-Huacuja

Math. Comput. Appl. 2021, 26(2), 28; https://doi.org/10.3390/mca26020028 - 31 Mar 2021

Cited by 2 | Viewed by 2215

Abstract

Many real-world optimization problems involving several conflicting objective functions frequently appear in current scenarios and it is expected they will remain present in the future. However, approaches combining multi-objective optimization with the incorporation of the decision maker’s (DM’s) preferences through multi-criteria ordinal classification [...] Read more.

Many real-world optimization problems involving several conflicting objective functions frequently appear in current scenarios and it is expected they will remain present in the future. However, approaches combining multi-objective optimization with the incorporation of the decision maker’s (DM’s) preferences through multi-criteria ordinal classification are still scarce. In addition, preferences are rarely associated with a DM’s characteristics; the preference selection is arbitrary. This paper proposes a new hybrid multi-objective optimization algorithm called P-HMCSGA (preference hybrid multi-criteria sorting genetic algorithm) that allows the DM’s preferences to be incorporated in the optimization process’ early phases and updated into the search process. P-HMCSGA incorporates preferences using a multi-criteria ordinal classification to distinguish solutions as good and bad; its parameters are determined with a preference disaggregation method. The main feature of P-HMCSGA is the new method proposed to associate preferences with the characterization profile of a DM and its integration with ordinal classification. This increases the selective pressure towards the desired region of interest more in agreement with the DM’s preferences specified in realistic profiles. The method is illustrated by solving real-size multi-objective PPPs (project portfolio problem). The experimentation aims to answer three questions: (i) To what extent does allowing the DM to express their preferences through a characterization profile impact the quality of the solution obtained in the optimization? (ii) How sensible is the proposal to different profiles? (iii) How much does the level of robustness of a profile impact the quality of final solutions (this question is related with the knowledge level that a DM has about his/her preferences)? Concluding, the proposal fulfills several desirable characteristics of a preferences incorporation method concerning these questions. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)

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14 pages, 403 KiB

Open AccessArticle

A Method for Integration of Preferences to a Multi-Objective Evolutionary Algorithm Using Ordinal Multi-Criteria Classification

by Alejandro Castellanos-Alvarez, Laura Cruz-Reyes, Eduardo Fernandez, Nelson Rangel-Valdez, Claudia Gómez-Santillán, Hector Fraire and José Alfredo Brambila-Hernández

Math. Comput. Appl. 2021, 26(2), 27; https://doi.org/10.3390/mca26020027 - 30 Mar 2021

Cited by 5 | Viewed by 2396

Abstract

Most real-world problems require the optimization of multiple objective functions simultaneously, which can conflict with each other. The environment of these problems usually involves imprecise information derived from inaccurate measurements or the variability in decision-makers’ (DMs’) judgments and beliefs, which can lead to [...] Read more.

Most real-world problems require the optimization of multiple objective functions simultaneously, which can conflict with each other. The environment of these problems usually involves imprecise information derived from inaccurate measurements or the variability in decision-makers’ (DMs’) judgments and beliefs, which can lead to unsatisfactory solutions. The imperfect knowledge can be present either in objective functions, restrictions, or decision-maker’s preferences. These optimization problems have been solved using various techniques such as multi-objective evolutionary algorithms (MOEAs). This paper proposes a new MOEA called NSGA-III-P (non-nominated sorting genetic algorithm III with preferences). The main characteristic of NSGA-III-P is an ordinal multi-criteria classification method for preference integration to guide the algorithm to the region of interest given by the decision-maker’s preferences. Besides, the use of interval analysis allows the expression of preferences with imprecision. The experiments contrasted several versions of the proposed method with the original NSGA-III to analyze different selective pressure induced by the DM’s preferences. In these experiments, the algorithms solved three-objectives instances of the DTLZ problem. The obtained results showed a better approximation to the region of interest for a DM when its preferences are considered. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)

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14 pages, 4017 KiB

Open AccessArticle

Investigation on the Mathematical Relation Model of Structural Reliability and Structural Robustness

by Qi-Wen Jin, Zheng Liu and Shuan-Hai He

Math. Comput. Appl. 2021, 26(2), 26; https://doi.org/10.3390/mca26020026 - 28 Mar 2021

Viewed by 2485

Abstract

Structural reliability and structural robustness, from different research fields, are usually employed for the evaluative analysis of building and civil engineering structures. Structural reliability has been widely used for structural analysis and optimization design, while structural robustness is still in rapid development. Several [...] Read more.

Structural reliability and structural robustness, from different research fields, are usually employed for the evaluative analysis of building and civil engineering structures. Structural reliability has been widely used for structural analysis and optimization design, while structural robustness is still in rapid development. Several dimensionless evaluation indexes have been defined for structural robustness so far, such as the structural reliability-based redundancy index. However, these different evaluation indexes are usually based on subjective definitions, and they are also difficult to put into engineering practice. The mathematical relational model between structural reliability and structural robustness has not been established yet. This paper is a quantitative study, focusing on the mathematical relation between structural reliability and structural robustness so as to further develop the theory of structural robustness. A strain energy evaluation index for structural robustness is introduced firstly by considering the energy principle. The mathematical relation model of structural reliability and structural robustness is then derived followed by a further comparative study on sensitivity, structural damage, and random variation factor. A cantilever beam and a truss beam are also presented as two case studies. In this study, a parabolic curve mathematical model between structural reliability and structural robustness is established. A significant variation trend for their sensitivities is also observed. The complex interaction mechanism of the joint effect of structural damage and random variation factor is also reflected. With consideration of the variation trend of the structural reliability index that is affected by different degrees of structural damage (mild impairment, moderate impairment, and severe impairment), a three-stage framework for structural life-cycle maintenance management is also proposed. This study can help us gain a better understanding of structural robustness and structural reliability. Some practical references are also provided for the better decision-making of maintenance and management departments. Full article

(This article belongs to the Special Issue Mathematical and Computational Modelling in Mechanics of Materials and Structures)

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21 pages, 468 KiB

Open AccessFeature PaperArticle

Impact of a New SARS-CoV-2 Variant on the Population: A Mathematical Modeling Approach

by Gilberto Gonzalez-Parra, David Martínez-Rodríguez and Rafael J. Villanueva-Micó

Math. Comput. Appl. 2021, 26(2), 25; https://doi.org/10.3390/mca26020025 - 27 Mar 2021

Cited by 35 | Viewed by 5260

Abstract

Several SARS-CoV-2 variants have emerged around the world, and the appearance of other variants depends on many factors. These new variants might have different characteristics that can affect the transmissibility and death rate. The administration of vaccines against the coronavirus disease 2019 (COVID-19) [...] Read more.

Several SARS-CoV-2 variants have emerged around the world, and the appearance of other variants depends on many factors. These new variants might have different characteristics that can affect the transmissibility and death rate. The administration of vaccines against the coronavirus disease 2019 (COVID-19) started in early December of 2020 and in some countries the vaccines will not soon be widely available. For this article, we studied the impact of a new more transmissible SARS-CoV-2 strain on prevalence, hospitalizations, and deaths related to the SARS-CoV-2 virus. We studied different scenarios regarding the transmissibility in order to provide a scientific support for public health policies and bring awareness of potential future situations related to the COVID-19 pandemic. We constructed a compartmental mathematical model based on differential equations to study these different scenarios. In this way, we are able to understand how a new, more infectious strain of the virus can impact the dynamics of the COVID-19 pandemic. We studied several metrics related to the possible outcomes of the COVID-19 pandemic in order to assess the impact of a higher transmissibility of a new SARS-CoV-2 strain on these metrics. We found that, even if the new variant has the same death rate, its high transmissibility can increase the number of infected people, those hospitalized, and deaths. The simulation results show that health institutions need to focus on increasing non-pharmaceutical interventions and the pace of vaccine inoculation since a new variant with higher transmissibility, such as, for example, VOC-202012/01 of lineage B.1.1.7, may cause more devastating outcomes in the population. Full article

(This article belongs to the Collection Mathematical Modelling of COVID-19)

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Math. Comput. Appl., Volume 26, Issue 2 (June 2021) – 24 articles

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