Mathematical and Computational Applications

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4 pages, 203 KiB

Open AccessEditorial

Preface to Numerical and Symbolic Computation: Developments and Applications—2021

by Maria Amélia R. Loja

Math. Comput. Appl. 2022, 27(6), 107; https://doi.org/10.3390/mca27060107 - 12 Dec 2022

Viewed by 1109

Abstract

This is the Special Issue “Numerical and Symbolic Computation: Developments and Applications—2021”, also available at the Special Issue website https://www [...] Full article

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

Research

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9 pages, 2939 KiB

Open AccessArticle

Increased Material Density within a New Biomechanism

by Carlos Aurelio Andreucci, Elza M. M. Fonseca and Renato N. Jorge

Math. Comput. Appl. 2022, 27(6), 90; https://doi.org/10.3390/mca27060090 - 2 Nov 2022

Cited by 9 | Viewed by 2387

Abstract

A new mechanism, applied in this study as a biomechanical device, known as a Bioactive Kinetic Screw (BKS) for bone implants is described. The BKS was designed as a bone implant, in which the bone particles, blood, cells, and protein molecules removed during [...] Read more.

A new mechanism, applied in this study as a biomechanical device, known as a Bioactive Kinetic Screw (BKS) for bone implants is described. The BKS was designed as a bone implant, in which the bone particles, blood, cells, and protein molecules removed during bone drilling are used as a homogeneous autogenous transplant at the same implant site, aiming to optimize the healing process and simplify the surgical procedure. In this work, the amount of bone that will be compacted inside and around the new biomechanism was studied, based on the density of the bone applied. This study allows us to analyze the average bone density in humans (1.85 mg/mm³ or 1850 µg/mm³) with four different synthetic bone densities (Sawbones PCF 10, 20, 30 and 40). The results show that across all four different synthetic bones densities, the bone within the new model is 3.45 times denser. After a pilot drill (with 10 mm length and 1.8 mm diameter), in cases where a guide hole is required, the increase in ratio is equal to 2.7 times inside and around the new biomechanism. The in vitro test validated the mathematical results, describing that in two different materials, the same compact factor of 3.45 was determined with the new biomechanical device. It was possible to describe that BKS can become a powerful tool in the diagnosis and treatment of natural bone conditions and any type of disease. Full article

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

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20 pages, 1768 KiB

Open AccessArticle

Comparison of Symbolic Computations for Solving Linear Delay Differential Equations Using the Laplace Transform Method

by Michelle Sherman, Gilbert Kerr and Gilberto González-Parra

Math. Comput. Appl. 2022, 27(5), 81; https://doi.org/10.3390/mca27050081 - 23 Sep 2022

Cited by 4 | Viewed by 2346

Abstract

In this paper, we focus on investigating the performance of the mathematical software program Maple and the programming language MATLAB when using these respective platforms to compute the method of steps (MoS) and the Laplace transform (LT) solutions for neutral and retarded linear [...] Read more.

In this paper, we focus on investigating the performance of the mathematical software program Maple and the programming language MATLAB when using these respective platforms to compute the method of steps (MoS) and the Laplace transform (LT) solutions for neutral and retarded linear delay differential equations (DDEs). We computed the analytical solutions that are obtained by using the Laplace transform method and the method of steps. The accuracy of the Laplace method solutions was determined (or assessed) by comparing them with those obtained by the method of steps. The Laplace transform method requires, among other mathematical tools, the use of the Cauchy residue theorem and the computation of an infinite series. Symbolic computation facilitates the whole process, providing solutions that would be unmanageable by hand. The results obtained here emphasize the fact that symbolic computation is a powerful tool for computing analytical solutions for linear delay differential equations. From a computational viewpoint, we found that the computation time is dependent on the complexity of the history function, the number of terms used in the LT solution, the number of intervals used in the MoS solution, and the parameters of the DDE. Finally, we found that, for linear non-neutral DDEs, MATLAB symbolic computations were faster than Maple. However, for linear neutral DDEs, which are often more complex to solve, Maple was faster. Regarding the accuracy of the LT solutions, Maple was, in a few cases, slightly better than MATLAB, but both were highly reliable. Full article

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

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24 pages, 2015 KiB

Open AccessArticle

Symbolic Computation Applied to Cauchy Type Singular Integrals

by Ana C. Conceição and Jéssica C. Pires

Math. Comput. Appl. 2022, 27(1), 3; https://doi.org/10.3390/mca27010003 - 31 Dec 2021

Cited by 3 | Viewed by 2723

Abstract

The development of operator theory is stimulated by the need to solve problems emerging from several fields in mathematics and physics. At the present time, this theory has wide applications in the study of non-linear differential equations, in linear transport theory, in the [...] Read more.

The development of operator theory is stimulated by the need to solve problems emerging from several fields in mathematics and physics. At the present time, this theory has wide applications in the study of non-linear differential equations, in linear transport theory, in the theory of diffraction of acoustic and electromagnetic waves, in the theory of scattering and of inverse scattering, among others. In our work, we use the computer algebra system Mathematica to implement, for the first time on a computer, analytical algorithms developed by us and others within operator theory. The main goal of this paper is to present new operator theory algorithms related to Cauchy type singular integrals, defined in the unit circle. The design of these algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. Several nontrivial examples computed with the algorithms are presented. The corresponding source code of the algorithms has been made available as a supplement to the online edition of this article. Full article

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

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19 pages, 2561 KiB

Open AccessArticle

Dynamic and Interactive Tools to Support Teaching and Learning

by Ana C. Conceição

Math. Comput. Appl. 2022, 27(1), 1; https://doi.org/10.3390/mca27010001 - 23 Dec 2021

Cited by 6 | Viewed by 3847

Abstract

The use of technological learning tools has been increasingly recognized as a useful tool to promote students’ motivation to deal with, and understand, mathematics concepts. Current digital technology allows students to work interactively with a large number and variety of graphics, complementing the [...] Read more.

The use of technological learning tools has been increasingly recognized as a useful tool to promote students’ motivation to deal with, and understand, mathematics concepts. Current digital technology allows students to work interactively with a large number and variety of graphics, complementing the theoretical results and often used paper and pencil calculations. The computer algebra system Mathematica is a very powerful software that allows the implementation of many interactive visual applications. The main goal of this work is to show how some new dynamic and interactive tools, created with Mathematica and available in the Computable Document Format (CDF), can be used as active learning tools to promote better student activity and engagement in the learning process. The CDF format allows anyone with a computer to use them, at no cost, even without an active Wolfram Mathematica license. Besides that, the presented tools are very intuitive to use which makes it suitable for less experienced users. Some tools applicable to several mathematics concepts taught in higher education will be presented. This kind of tools can be used either in a remote or classroom learning environment. The corresponding CDF files are made available as supplement of the online edition of this article. Full article

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

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17 pages, 1023 KiB

Open AccessEditor’s ChoiceArticle

Modelling Forest Fires Using Complex Networks

by Sara Perestrelo, Maria C. Grácio, Nuno A. Ribeiro and Luís M. Lopes

Math. Comput. Appl. 2021, 26(4), 68; https://doi.org/10.3390/mca26040068 - 28 Sep 2021

Cited by 3 | Viewed by 3139

Abstract

Forest fires have been a major threat to the environment throughout history. In order to mitigate its consequences, we present, in a first of a series of works, a mathematical model with the purpose of predicting fire spreading in a given land portion [...] Read more.

Forest fires have been a major threat to the environment throughout history. In order to mitigate its consequences, we present, in a first of a series of works, a mathematical model with the purpose of predicting fire spreading in a given land portion divided into patches, considering the area and the rate of spread of each patch as inputs. The rate of spread can be estimated from previous knowledge on fuel availability, weather and terrain conditions. We compute the time duration of the spreading process in a land patch in order to construct and parametrize a landscape network, using cellular automata simulations. We use the multilayer network model to propose a network of networks at the landscape scale, where the nodes are the local patches, each with their own spreading dynamics. We compute some respective network measures and aim, in further work, for the establishment of a fire-break structure according to increasing accuracy simulation results. Full article

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

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10 pages, 307 KiB

Open AccessArticle

Using the Evolution Operator to Classify Evolution Algebras

by Desamparados Fernández-Ternero, Víctor M. Gómez-Sousa and Juan Núñez-Valdés

Math. Comput. Appl. 2021, 26(3), 57; https://doi.org/10.3390/mca26030057 - 5 Aug 2021

Cited by 2 | Viewed by 1914

Abstract

Evolution algebras are currently widely studied due to their importance not only “per se” but also for their many applications to different scientific disciplines, such as Physics or Engineering, for instance. This paper deals with these types of algebras and their applications. A [...] Read more.

Evolution algebras are currently widely studied due to their importance not only “per se” but also for their many applications to different scientific disciplines, such as Physics or Engineering, for instance. This paper deals with these types of algebras and their applications. A criterion for classifying those satisfying certain conditions is given and an algorithm to obtain degenerate evolution algebras starting from those of smaller dimensions is also analyzed and constructed. Full article

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

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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 3525

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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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 3437

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, 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 3104

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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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 4764

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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25 pages, 1906 KiB

Open AccessArticle

A Framework for Analysis and Prediction of Operational Risk Stress

by Peter Mitic

Math. Comput. Appl. 2021, 26(1), 19; https://doi.org/10.3390/mca26010019 - 24 Feb 2021

Cited by 3 | Viewed by 2875

Abstract

A model for financial stress testing and stability analysis is presented. Given operational risk loss data within a time window, short-term projections are made using Loess fits to sequences of lognormal parameters. The projections can be scaled by a sequence of risk factors, [...] Read more.

A model for financial stress testing and stability analysis is presented. Given operational risk loss data within a time window, short-term projections are made using Loess fits to sequences of lognormal parameters. The projections can be scaled by a sequence of risk factors, derived from economic data in response to international regulatory requirements. Historic and projected loss data are combined using a lengthy nonlinear algorithm to calculate a capital reserve for the upcoming year. The model is embedded in a general framework, in which arrays of risk factors can be swapped in and out to assess their effect on the projected losses. Risk factor scaling is varied to assess the resilience and stability of financial institutions to economic shock. Symbolic analysis of projected losses shows that they are well-conditioned with respect to risk factors. Specific reference is made to the effect of the 2020 COVID-19 pandemic. For a 1-year projection, the framework indicates a requirement for an increase in regulatory capital of approximately 3% for mild stress, 8% for moderate stress, and 32% for extreme stress. The proposed framework is significant because it is the first formal methodology to link financial risk with economic factors in an objective way without recourse to correlations. Full article

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

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