Mathematical and Computational Applications

9 pages, 634 KiB

Open AccessArticle

Exact Evaluation of Statistical Moments in Superradiant Emission

by Gilberto M. Nakamura, Brenno Cabella and Alexandre S. Martinez

Math. Comput. Appl. 2019, 24(2), 66; https://doi.org/10.3390/mca24020066 - 23 Jun 2019

Viewed by 2738

Superradiance describes the coherent collective radiation caused by the interaction between many emitters, mediated by a shared electromagnetic field. Recent experiments involving Bose–Einstein condensates coupled to high-finesse cavities and interacting quantum dots in condensed-matter have attracted attention to the superradiant regime as a [...] Read more.

Superradiance describes the coherent collective radiation caused by the interaction between many emitters, mediated by a shared electromagnetic field. Recent experiments involving Bose–Einstein condensates coupled to high-finesse cavities and interacting quantum dots in condensed-matter have attracted attention to the superradiant regime as a fundamental step to create quantum technologies. Here, we consider a simplified description of superradiance that allows the evaluation of statistical moments. A correspondence with the classical birthday problem recovers the statistical moments for discrete time and an arbitrary number of emitters. In addition, the correspondence provides a way to calculate the degeneracy of the problem. Full article

(This article belongs to the Special Issue Dynamics Days Latin America and the Caribbean 2018)

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16 pages, 7897 KiB

Open AccessArticle

Mathematical Models for Simulation and Optimization of High-Flux Solar Furnaces

by José Carlos Garcia Pereira, Jorge Cruz Fernandes and Luís Guerra Rosa

Math. Comput. Appl. 2019, 24(2), 65; https://doi.org/10.3390/mca24020065 - 21 Jun 2019

Cited by 5 | Viewed by 4365

Abstract

High-flux solar furnaces distributed throughout the world have been designed and constructed individually, i.e., on a one-by-one basis because there are several possible optical configurations that must take into account the geographical location and the maximum power to be attained. In this work, [...] Read more.

High-flux solar furnaces distributed throughout the world have been designed and constructed individually, i.e., on a one-by-one basis because there are several possible optical configurations that must take into account the geographical location and the maximum power to be attained. In this work, three ray-tracing models were developed to simulate the optical paths travelled by sun rays in solar furnaces of high concentration using as an example, the solar furnace SF60 of the Plataforma Solar de Almería, in Spain. All these simulation models are supported by mathematical constructions, which are also presented. The first model assumes a random distribution of sun rays coming from a concentrator with spherical curvature. The second model assumes that a random distribution of parallel rays coming from the heliostat is reflected by a concentrator with spherical curvature. Finally, the third model considers that the random parallel rays are reflected by a concentrator with a paraboloid curvature. The three models are all important in optical geometry, although the paraboloid model is more adequate to optimize solar furnaces. The models are illustrated by studying the influence of mirror positioning and shutter attenuation. Additionally, ray-tracing simulations confirmed the possibility to attain homogenous distribution of flux by means of double reflexion using two paraboloid surfaces. Full article

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17 pages, 462 KiB

Open AccessArticle

Bisimulation for Secure Information Flow Analysis of Multi-Threaded Programs

by Ali A. Noroozi, Jaber Karimpour and Ayaz Isazadeh

Math. Comput. Appl. 2019, 24(2), 64; https://doi.org/10.3390/mca24020064 - 17 Jun 2019

Cited by 4 | Viewed by 3380

Abstract

Preserving the confidentiality of information is a growing concern in software development. Secure information flow is intended to maintain the confidentiality of sensitive information by preventing them from flowing to attackers. This paper discusses how to ensure confidentiality for multi-threaded programs through a [...] Read more.

Preserving the confidentiality of information is a growing concern in software development. Secure information flow is intended to maintain the confidentiality of sensitive information by preventing them from flowing to attackers. This paper discusses how to ensure confidentiality for multi-threaded programs through a property called observational determinism. Operational semantics of multi-threaded programs are modeled using Kripke structures. Observational determinism is formalized in terms of divergence weak low-bisimulation. Bisimulation is an equivalence relation associating executions that simulate each other. The new property is called bisimulation-based observational determinism. Furthermore, a model checking method is proposed to verify the new property and ensure that secure information flow holds in a multi-threaded program. The model checking method successively refines the Kripke model of the program until the quotient of the model with respect to divergence weak low-bisimulation is reached. Then, bisimulation-based observational determinism is checked on the quotient, which is a minimized model of the concrete Kripke model. The time complexity of the proposed method is polynomial in the size of the Kripke model. The proposed approach has been implemented on top of PRISM, a probabilistic model checking tool. Finally, a case study is discussed to show the applicability of the proposed approach. Full article

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16 pages, 1119 KiB

Open AccessArticle

Analysis of a Model of Leishmaniasis with Multiple Time Lags in All Populations

by Ephraim Agyingi and Tamas Wiandt

Math. Comput. Appl. 2019, 24(2), 63; https://doi.org/10.3390/mca24020063 - 15 Jun 2019

Cited by 3 | Viewed by 2863

Abstract

There are several types of deterministic compartmental models for disease epidemiology such as SIR, SIS, SEIS, SEIR, etc. The exposed population group in, for example SEIS or SEIR, usually represents individuals in the incubation class. Time delays (of which there are several types) [...] Read more.

There are several types of deterministic compartmental models for disease epidemiology such as SIR, SIS, SEIS, SEIR, etc. The exposed population group in, for example SEIS or SEIR, usually represents individuals in the incubation class. Time delays (of which there are several types) when incorporated into a SIR or SIS model, also fulfil the role of the incubation period without necessarily adding another compartment to the model. This paper incorporates time delays into a SIS model that describes the transmission dynamics of cutaneous leishmaniasis. The time lags account for the incubation periods within the sandflies vector, the human hosts and the different animal groups that serve as reservoir hosts. A threshold value,

R_{0}

, of the model is computed and used to study the disease-free equilibrium and endemic equilibrium of the system. Analysis demonstrating local and global stability of the disease-free equilibrium when

R_{0} < 1

is provided for all

n + 1

population groups involved is provided. The existence of an endemic equilibrium is only guaranteed when

R_{0} > 1

and numerical analysis of the endemic equilibrium for a human host, a vector host and a single animal reservoir host that is globally stable is also provided. Full article

(This article belongs to the Section Natural Sciences)

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13 pages, 278 KiB

Open AccessArticle

A Fast Factorisation of Semi-Primes Using Sum of Squares

by Anthony Overmars and Sitalakshmi Venkatraman

Math. Comput. Appl. 2019, 24(2), 62; https://doi.org/10.3390/mca24020062 - 11 Jun 2019

Cited by 5 | Viewed by 4111

Abstract

For several centuries, prime factorisation of large numbers has drawn much attention due its practical applications and the associated challenges. In computing applications, encryption algorithms such as the Rivest–Shamir–Adleman (RSA) cryptosystems are widely used for information security, where the keys (public and private) [...] Read more.

For several centuries, prime factorisation of large numbers has drawn much attention due its practical applications and the associated challenges. In computing applications, encryption algorithms such as the Rivest–Shamir–Adleman (RSA) cryptosystems are widely used for information security, where the keys (public and private) of the encryption code are represented using large prime factors. Since prime factorisation of large numbers is extremely hard, RSA cryptosystems take advantage of this property to ensure information security. A semi-prime being, a product of two prime numbers, has wide applications in RSA algorithms and pseudo number generators. In this paper, we consider a semi-prime number whose construction consists of primes,

N = p_{1} p_{2}

, being Pythagorean and having a representation on the Cartesian plane such that,

p = x^{2} + y^{2}

. We prove that the product of two such primes can be represented as the sum of four squares, and further, that the sums of two squares can be derived. For such a semi-prime, if the original construction is unknown and the sum of four squares is known, by Euler’s factorisation the original construction

p_{1} p_{2}

can be found. By considering the parity of each of the squares, we propose a new method of factorisation of semi-primes. Our factorisation method provides a faster alternative to Euler’s method by exploiting the relationship between the four squares. The correctness of the new factorisation method is established with mathematical proofs and its practical value is demonstrated by generating RSA-768 efficiently. Full article

14 pages, 783 KiB

Open AccessArticle

Functional Ca²⁺ Channels between Channel Clusters are Necessary for the Propagation of IP₃R-Mediated Ca²⁺ Waves

by Estefanía Piegari and Silvina Ponce Dawson

Math. Comput. Appl. 2019, 24(2), 61; https://doi.org/10.3390/mca24020061 - 11 Jun 2019

Cited by 1 | Viewed by 2375

Abstract

The specificity and universality of intracellular Ca

^{2 +}

signals rely on the variety of spatio-temporal patterns that the Ca

^{2 +}

concentration can display. Ca

^{2 +}

release into the cytosol through inositol 1,4,5-trisphosphate receptors (IP

_{3}

Rs) is key for this [...] Read more.

The specificity and universality of intracellular Ca

^{2 +}

signals rely on the variety of spatio-temporal patterns that the Ca

^{2 +}

concentration can display. Ca

^{2 +}

release into the cytosol through inositol 1,4,5-trisphosphate receptors (IP

_{3}

Rs) is key for this variety. The opening probability of IP

_{3}

Rs depends on the cytosolic Ca

^{2 +}

concentration. All of the dynamics are then well described by an excitable system in which the signal propagation depends on the ability of the Ca

^{2 +}

released through one IP

_{3}

R to induce the opening of other IP

_{3}

Rs. In most cell types, IP

_{3}

Rs are organized in clusters, i.e., the cytosol is a “patchy” excitable system in which the signals can remain localized (i.e., involving the release through one or more IP

_{3}

Rs in a cluster), or become global depending on the efficiency of the Ca

^{2 +}

-mediated coupling between clusters. The spatial range over which the signals propagate determines the responses that the cell eventually produces. This points to the importance of understanding the mechanisms that make the propagation possible. Our previous qualitative comparison between experiments and numerical simulations seemed to indicate that Ca

^{2 +}

release not only occurs within the close vicinity of the clearly identifiable release sites (IP

_{3}

R clusters) but that there are also functional IP

_{3}

Rs in between them. In this paper, we present a quantitative comparison between experiments and models that corroborate this preliminary conclusion. This result has implications on how the Ca

^{2 +}

-mediated coupling between clusters works and how it can eventually be disrupted by the different Ca

^{2 +}

trapping mechanisms. Full article

(This article belongs to the Special Issue Dynamics Days Latin America and the Caribbean 2018)

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22 pages, 7810 KiB

Open AccessFeature PaperArticle

Combining the Magnetic Equivalent Circuit and Maxwell–Fourier Method for Eddy-Current Loss Calculation

by Youcef Benmessaoud, Frédéric Dubas and Mickael Hilairet

Math. Comput. Appl. 2019, 24(2), 60; https://doi.org/10.3390/mca24020060 - 4 Jun 2019

Cited by 7 | Viewed by 4052

Abstract

In this paper, a hybrid model in Cartesian coordinates combining a two-dimensional (2-D) generic magnetic equivalent circuit (MEC) with a 2-D analytical model based on the Maxwell–Fourier method (i.e., the formal resolution of Maxwell’s equations by using the separation of variables method and [...] Read more.

In this paper, a hybrid model in Cartesian coordinates combining a two-dimensional (2-D) generic magnetic equivalent circuit (MEC) with a 2-D analytical model based on the Maxwell–Fourier method (i.e., the formal resolution of Maxwell’s equations by using the separation of variables method and the Fourier’s series) is developed. This model coupling has been applied to a U-cored static electromagnetic device. The main objective is to compute the magnetic field behavior in massive conductive parts (e.g., aluminum, magnets, copper, iron) considering the skin effect (i.e., with the eddy-current reaction field) and to predict the eddy-current losses. The magnetic field distribution for various models is validated with 2-D and three-dimensional (3-D) finite-element analysis (FEA). The study is also focused on the discretization influence of 2-D generic MEC on the eddy-current loss calculation in conductive regions. Experimental tests and 3-D FEA have been compared with the proposed approach on massive conductive parts in aluminum. For an operating point, the computation time is divided by ~4.6 with respect to 3-D FEA. Full article

(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)

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16 pages, 837 KiB

Open AccessArticle

Some New Higher Order Weighted Newton Methods for Solving Nonlinear Equation with Applications

by Parimala Sivakumar and Jayakumar Jayaraman

Math. Comput. Appl. 2019, 24(2), 59; https://doi.org/10.3390/mca24020059 - 3 Jun 2019

Cited by 8 | Viewed by 2964

Abstract

Three new iterative methods for solving scalar nonlinear equations using weight function technique are presented. The first one is a two-step fifth order method with four function evaluations which is improved from a two-step Newton’s method having same number of function evaluations. By [...] Read more.

Three new iterative methods for solving scalar nonlinear equations using weight function technique are presented. The first one is a two-step fifth order method with four function evaluations which is improved from a two-step Newton’s method having same number of function evaluations. By this, the efficiency index of the new method is improved from 1.414 to 1.495. The second one is a three step method with one additional function evaluation producing eighth order accuracy with efficiency index 1.516. The last one is a new fourth order optimal two-step method with efficiency index 1.587. All these three methods are better than Newton’s method and many other equivalent higher order methods. Convergence analyses are established so that these methods have fifth, eighth and fourth order respectively. Numerical examples ascertain that the proposed methods are efficient and demonstrate better performance when compared to some equivalent and optimal methods. Seven application problems are solved to illustrate the efficiency and performance of the proposed methods. Full article

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7 pages, 258 KiB

Open AccessCommunication

Mixed Boundary Value Problems for the Elasticity System in Exterior Domains

by Hovik A. Matevossian

Math. Comput. Appl. 2019, 24(2), 58; https://doi.org/10.3390/mca24020058 - 2 Jun 2019

Cited by 2 | Viewed by 2540

Abstract

We study the properties of solutions of the mixed Dirichlet–Robin and Neumann–Robin problems for the linear system of elasticity theory in the exterior of a compact set and the asymptotic behavior of solutions of these problems at infinity under the assumption that the [...] Read more.

We study the properties of solutions of the mixed Dirichlet–Robin and Neumann–Robin problems for the linear system of elasticity theory in the exterior of a compact set and the asymptotic behavior of solutions of these problems at infinity under the assumption that the energy integral with weight

{| x |}^{a}

is finite for such solutions. We use the variational principle and depending on the value of the parameter a, obtain uniqueness (non-uniqueness) theorems of the mixed problems or present exact formulas for the dimension of the space of solutions. Full article

(This article belongs to the Special Issue Related Problems of Continuum Mechanics)

27 pages, 4128 KiB

Open AccessFeature PaperArticle

Data-Driven Microstructure Property Relations

by Julian Lißner and Felix Fritzen

Math. Comput. Appl. 2019, 24(2), 57; https://doi.org/10.3390/mca24020057 - 31 May 2019

Cited by 5 | Viewed by 3995

Abstract

An image based prediction of the effective heat conductivity for highly heterogeneous microstructured materials is presented. The synthetic materials under consideration show different inclusion morphology, orientation, volume fraction and topology. The prediction of the effective property is made exclusively based on image data [...] Read more.

An image based prediction of the effective heat conductivity for highly heterogeneous microstructured materials is presented. The synthetic materials under consideration show different inclusion morphology, orientation, volume fraction and topology. The prediction of the effective property is made exclusively based on image data with the main emphasis being put on the 2-point spatial correlation function. This task is implemented using both unsupervised and supervised machine learning methods. First, a snapshot proper orthogonal decomposition (POD) is used to analyze big sets of random microstructures and, thereafter, to compress significant characteristics of the microstructure into a low-dimensional feature vector. In order to manage the related amount of data and computations, three different incremental snapshot POD methods are proposed. In the second step, the obtained feature vector is used to predict the effective material property by using feed forward neural networks. Numerical examples regarding the incremental basis identification and the prediction accuracy of the approach are presented. A Python code illustrating the application of the surrogate is freely available. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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28 pages, 1750 KiB

Open AccessFeature PaperArticle

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

by Oliver Kunc and Felix Fritzen

Math. Comput. Appl. 2019, 24(2), 56; https://doi.org/10.3390/mca24020056 - 27 May 2019

Cited by 18 | Viewed by 3771 | Correction

Abstract

The computational homogenization of hyperelastic solids in the geometrically nonlinear context has yet to be treated with sufficient efficiency in order to allow for real-world applications in true multiscale settings. This problem is addressed by a problem-specific surrogate model founded on a reduced [...] Read more.

The computational homogenization of hyperelastic solids in the geometrically nonlinear context has yet to be treated with sufficient efficiency in order to allow for real-world applications in true multiscale settings. This problem is addressed by a problem-specific surrogate model founded on a reduced basis approximation of the deformation gradient on the microscale. The setup phase is based upon a snapshot POD on deformation gradient fluctuations, in contrast to the widespread displacement-based approach. In order to reduce the computational offline costs, the space of relevant macroscopic stretch tensors is sampled efficiently by employing the Hencky strain. Numerical results show speed-up factors in the order of 5–100 and significantly improved robustness while retaining good accuracy. An open-source demonstrator tool with 50 lines of code emphasizes the simplicity and efficiency of the method. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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18 pages, 320 KiB

Open AccessArticle

Augmented Lagrangian Approach to the Newsvendor Model with Component Commonality

by Abdelouahed Hamdi and Lotfi Tadj

Math. Comput. Appl. 2019, 24(2), 55; https://doi.org/10.3390/mca24020055 - 24 May 2019

Viewed by 2563

Abstract

Component commonality is a well-known approach in manufacturing, where the same components are used for multiple products. It has been implemented by many established companies such as Airbus, Kodak, Toyota, etc. We consider a standard two-product inventory model with a common component. The [...] Read more.

Component commonality is a well-known approach in manufacturing, where the same components are used for multiple products. It has been implemented by many established companies such as Airbus, Kodak, Toyota, etc. We consider a standard two-product inventory model with a common component. The demands for the products are independent random variables. Instead of the usual approach to minimize the total shortage quantity, we propose to minimize the total shortage cost. The resulting problem is a non-convex nonlinear mathematical program. We illustrate the use of a primal-dual proximal method to solve this problem by obtaining numerically the optimal allocations of components. In particular, we show that a higher unit shortage cost induces a higher allocation. Full article

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11 pages, 1016 KiB

Open AccessArticle

On the Approximation of a Nonlinear Biological Population Model Using Localized Radial Basis Function Method

by Marjan Uddin, Hazrat Ali and Muhammad Taufiq

Math. Comput. Appl. 2019, 24(2), 54; https://doi.org/10.3390/mca24020054 - 23 May 2019

Cited by 5 | Viewed by 2673

Abstract

A localized radial basis function meshless method is applied to approximate a nonlinear biological population model with highly satisfactory results. The method approximates the derivatives at every point corresponding to their local support domain. The method is well suited for arbitrary domains. Compared [...] Read more.

A localized radial basis function meshless method is applied to approximate a nonlinear biological population model with highly satisfactory results. The method approximates the derivatives at every point corresponding to their local support domain. The method is well suited for arbitrary domains. Compared to the finite element and element free Galerkin methods, no integration tool is required. Four examples are demonstrated to check the efficiency and accuracy of the method. The results are compared with an exact solution and other methods available in literature. Full article

(This article belongs to the Special Issue Computational Methods in Interdisciplinary Applications of Nonlinear Dynamics)

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20 pages, 323 KiB

Open AccessArticle

A Study on Novel Extensions for the p-adic Gamma and p-adic Beta Functions

by Ugur Duran and Mehmet Acikgoz

Math. Comput. Appl. 2019, 24(2), 53; https://doi.org/10.3390/mca24020053 - 21 May 2019

Cited by 1 | Viewed by 2396

Abstract

In this paper, we introduce the

(ρ, q)

-analog of the p-adic factorial function. By utilizing some properties of

(ρ, q)

-numbers, we obtain several new and interesting identities and formulas. We then construct the p-adic [...] Read more.

In this paper, we introduce the

(ρ, q)

-analog of the p-adic factorial function. By utilizing some properties of

(ρ, q)

-numbers, we obtain several new and interesting identities and formulas. We then construct the p-adic

(ρ, q)

-gamma function by means of the mentioned factorial function. We investigate several properties and relationships belonging to the foregoing gamma function, some of which are given for the case

p = 2

. We also derive more representations of the p-adic

(ρ, q)

-gamma function in general case. Moreover, we consider the p-adic

(ρ, q)

-Euler constant derived from the derivation of p-adic

(ρ, q)

-gamma function at

x = 1

. Furthermore, we provide a limit representation of aforementioned Euler constant based on

(ρ, q)

-numbers. Finally, we consider

(ρ, q)

-extension of the p-adic beta function via the p-adic

(ρ, q)

-gamma function and we then investigate various formulas and identities. Full article

(This article belongs to the Section Natural Sciences)

21 pages, 3930 KiB

Open AccessFeature PaperArticle

Natural Frequency Analysis of Functionally Graded Orthotropic Cross-Ply Plates Based on the Finite Element Method

by Michele Bacciocchi and Angelo Marcello Tarantino

Math. Comput. Appl. 2019, 24(2), 52; https://doi.org/10.3390/mca24020052 - 19 May 2019

Cited by 16 | Viewed by 3370

Abstract

This paper aims to present a finite element (FE) formulation for the study of the natural frequencies of functionally graded orthotropic laminated plates characterized by cross-ply layups. A nine-node Lagrange element is considered for this purpose. The main novelty of the research is [...] Read more.

This paper aims to present a finite element (FE) formulation for the study of the natural frequencies of functionally graded orthotropic laminated plates characterized by cross-ply layups. A nine-node Lagrange element is considered for this purpose. The main novelty of the research is the modelling of the reinforcing fibers of the orthotropic layers assuming a non-uniform distribution in the thickness direction. The Halpin–Tsai approach is employed to define the overall mechanical properties of the composite layers starting from the features of the two constituents (fiber and epoxy resin). Several functions are introduced to describe the dependency on the thickness coordinate of their volume fraction. The analyses are carried out in the theoretical framework provided by the first-order shear deformation theory (FSDT) for laminated thick plates. Nevertheless, the same approach is used to deal with the vibration analysis of thin plates, neglecting the shear stiffness of the structure. This objective is achieved by properly choosing the value of the shear correction factor, without any modification in the formulation. The results prove that the dynamic response of thin and thick plates, in terms of natural frequencies and mode shapes, is affected by the non-uniform placement of the fibers along the thickness direction. Full article

(This article belongs to the Special Issue Mesh-Free and Finite Element-Based Methods for Structural Mechanics Applications)

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16 pages, 2553 KiB

Open AccessFeature PaperArticle

Structures and Instabilities in Reaction Fronts Separating Fluids of Different Densities

by Johan Llamoza and Desiderio A. Vasquez

Math. Comput. Appl. 2019, 24(2), 51; https://doi.org/10.3390/mca24020051 - 17 May 2019

Viewed by 2643

Abstract

Density gradients across reaction fronts propagating vertically can lead to Rayleigh–Taylor instabilities. Reaction fronts can also become unstable due to diffusive instabilities, regardless the presence of a mass density gradient. In this paper, we study the interaction between density driven convection and fronts [...] Read more.

Density gradients across reaction fronts propagating vertically can lead to Rayleigh–Taylor instabilities. Reaction fronts can also become unstable due to diffusive instabilities, regardless the presence of a mass density gradient. In this paper, we study the interaction between density driven convection and fronts with diffusive instabilities. We focus in fluids confined in Hele–Shaw cells or porous media, with the hydrodynamics modeled by Brinkman’s equation. The time evolution of the front is described with a Kuramoto–Sivashinsky (KS) equation coupled to the fluid velocity. A linear stability analysis shows a transition to convection that depends on the density differences between reacted and unreacted fluids. A stabilizing density gradient can surpress the effects of diffusive instabilities. The two-dimensional numerical solutions of the nonlinear equations show an increase of speed due to convection. Brinkman’s equation lead to the same results as Darcy’s laws for narrow gap Hele–Shaw cells. For large gaps, modeling the hydrodynamics using Stokes’ flow lead to the same results. Full article

(This article belongs to the Special Issue Dynamics Days Latin America and the Caribbean 2018)

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16 pages, 4616 KiB

Open AccessArticle

Time Recurrence Analysis of a Near Singular Billiard

by Rodrigo Simile Baroni, Ricardo Egydio de Carvalho, Bruno Castaldi and Bruno Furlanetto

Math. Comput. Appl. 2019, 24(2), 50; https://doi.org/10.3390/mca24020050 - 8 May 2019

Cited by 1 | Viewed by 3000

Abstract

Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like. We call this configuration [...] Read more.

Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like. We call this configuration the near singular, in which a single initial condition (IC) densely fills the phase space with straight lines. To characterize the orbits, two techniques were applied: (i) Finite-time Lyapunov exponent (FTLE) and (ii) time recurrence. The largest Lyapunov exponent λ was calculated using the FTLE method, which for conservative systems, λ > 0 indicates chaotic behavior and λ = 0 indicates regularity. The recurrence of orbits in the phase space was investigated through recurrence plots. Chaotic orbits show many different return times and, according to Slater’s theorem, quasi-periodic orbits have at most three different return times, the bigger one being the sum of the other two. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of λ, suggesting some change in the dynamical behavior of the system. Many different recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. We also conclude that reducing the magnitude of the nonlinear part of the system did not prevent chaotic behavior. Full article

(This article belongs to the Special Issue Dynamics Days Latin America and the Caribbean 2018)

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16 pages, 322 KiB

Open AccessArticle

A Unified Generalization of the Catalan, Fuss, and Fuss—Catalan Numbers

by Feng Qi, Xiao-Ting Shi and Pietro Cerone

Math. Comput. Appl. 2019, 24(2), 49; https://doi.org/10.3390/mca24020049 - 8 May 2019

Cited by 7 | Viewed by 4101

Abstract

In the paper, the authors introduce a unified generalization of the Catalan numbers, the Fuss numbers, the Fuss–Catalan numbers, and the Catalan–Qi function, and discover some properties of the unified generalization, including a product-ratio expression of the unified generalization in terms of the [...] Read more.

In the paper, the authors introduce a unified generalization of the Catalan numbers, the Fuss numbers, the Fuss–Catalan numbers, and the Catalan–Qi function, and discover some properties of the unified generalization, including a product-ratio expression of the unified generalization in terms of the Catalan–Qi functions, three integral representations of the unified generalization, and the logarithmically complete monotonicity of the second order for a special case of the unified generalization. Full article

7 pages, 615 KiB

Open AccessFeature PaperArticle

Inadequate Sampling Rates Can Undermine the Reliability of Ecological Interaction Estimation

by Brenno Cabella, Fernando Meloni and Alexandre S. Martinez

Math. Comput. Appl. 2019, 24(2), 48; https://doi.org/10.3390/mca24020048 - 30 Apr 2019

Cited by 4 | Viewed by 3891

Abstract

Cycles in population dynamics are abundant in nature and are understood as emerging from the interaction among coupled species. When sampling is conducted at a slow rate compared to the population cycle period (aliasing effect), one is prone to misinterpretations. However, aliasing has [...] Read more.

Cycles in population dynamics are abundant in nature and are understood as emerging from the interaction among coupled species. When sampling is conducted at a slow rate compared to the population cycle period (aliasing effect), one is prone to misinterpretations. However, aliasing has been poorly addressed in coupled population dynamics. To illustrate the aliasing effect, the Lotka–Volterra model oscillatory regime is numerically sampled, creating prey–predator cycles. We show that inadequate sampling rates may produce inversions in the cause-effect relationship among other artifacts. More generally, slow acquisition rates may distort data interpretation and produce deceptive patterns and eventually leading to misinterpretations, as predators becoming preys. Experiments in coupled population dynamics should be designed that address the eventual aliasing effect. Full article

(This article belongs to the Special Issue Dynamics Days Latin America and the Caribbean 2018)

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20 pages, 7960 KiB

Open AccessArticle

A Transdisciplinary Approach for Analyzing Stress Flow Patterns in Biostructures

by Reena Patel, Guillermo Riveros, David Thompson, Edward Perkins, Jan Jeffery Hoover, John Peters and Antoinette Tordesillas

Math. Comput. Appl. 2019, 24(2), 47; https://doi.org/10.3390/mca24020047 - 26 Apr 2019

Cited by 4 | Viewed by 3964

Abstract

This work presents a transdisciplinary, integrated approach that uses computational mechanics experiments with a flow network strategy to gain fundamental insights into the stress flow of high-performance, lightweight, structured composites by investigating the rostrum of paddlefish. Although computational mechanics experiments give an overall [...] Read more.

This work presents a transdisciplinary, integrated approach that uses computational mechanics experiments with a flow network strategy to gain fundamental insights into the stress flow of high-performance, lightweight, structured composites by investigating the rostrum of paddlefish. Although computational mechanics experiments give an overall distribution of stress in the structural systems, stress flow patterns formed at nascent stages of loading a biostructure are hard to determine. Computational mechanics experiments on a complex model will involve a high degree of freedom thereby making the extraction of finer details computationally expensive. To address this challenge, the evolution of the stress in the rostrum is formulated as a network flow problem generated by extracting the node and connectivity information from the numerical model of the rostrum. The flow network is weighted based on the parameter of interest, which is stress in the current research. The changing kinematics of the system is provided as input to the mathematical algorithm that computes the minimum cut of the flow network. The flow network approach is verified using two simple classical problems. When applied to the model of the rostrum, the flow network approach identifies strain localization in tensile regions, and buckling/crushing in compressive regions. Full article

(This article belongs to the Special Issue Mesh-Free and Finite Element-Based Methods for Structural Mechanics Applications)

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12 pages, 6084 KiB

Open AccessArticle

Suppression of Phase Synchronization in Scale-Free Neural Networks Using External Pulsed Current Protocols

by Bruno Rafael Reichert Boaretto, Roberto C. Budzinski, Thiago L. Prado and Sergio Roberto Lopes

Math. Comput. Appl. 2019, 24(2), 46; https://doi.org/10.3390/mca24020046 - 24 Apr 2019

Cited by 2 | Viewed by 3039

Abstract

The synchronization of neurons is fundamental for the functioning of the brain since its lack or excess may be related to neurological disorders, such as autism, Parkinson’s and neuropathies such as epilepsy. In this way, the study of synchronization, as well as its [...] Read more.

The synchronization of neurons is fundamental for the functioning of the brain since its lack or excess may be related to neurological disorders, such as autism, Parkinson’s and neuropathies such as epilepsy. In this way, the study of synchronization, as well as its suppression in coupled neurons systems, consists of an important multidisciplinary research field where there are still questions to be answered. Here, through mathematical modeling and numerical approach, we simulated a neural network composed of 5000 bursting neurons in a scale-free connection scheme where non-trivial synchronization phenomenon is observed. We proposed two different protocols to the suppression of phase synchronization, which is related to deep brain stimulation and delayed feedback control. Through an optimization process, it is possible to suppression the abnormal synchronization in the neural network. Full article

(This article belongs to the Special Issue Dynamics Days Latin America and the Caribbean 2018)

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21 pages, 21203 KiB

Open AccessFeature PaperArticle

Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations

by Nissrine Akkari, Fabien Casenave and Vincent Moureau

Math. Comput. Appl. 2019, 24(2), 45; https://doi.org/10.3390/mca24020045 - 24 Apr 2019

Cited by 13 | Viewed by 4396

Abstract

In the following paper, we consider the problem of constructing a time stable reduced order model of the 3D turbulent and incompressible Navier–Stokes equations. The lack of stability associated with the order reduction methods of the Navier–Stokes equations is a well-known problem and, [...] Read more.

In the following paper, we consider the problem of constructing a time stable reduced order model of the 3D turbulent and incompressible Navier–Stokes equations. The lack of stability associated with the order reduction methods of the Navier–Stokes equations is a well-known problem and, in general, it is very difficult to account for different scales of a turbulent flow in the same reduced space. To remedy this problem, we propose a new stabilization technique based on an a priori enrichment of the classical proper orthogonal decomposition (POD) modes with dissipative modes associated with the gradient of the velocity fields. The main idea is to be able to do an a priori analysis of different modes in order to arrange a POD basis in a different way, which is defined by the enforcement of the energetic dissipative modes within the first orders of the reduced order basis. This enables us to model the production and the dissipation of the turbulent kinetic energy (TKE) in a separate fashion within the high ranked new velocity modes, hence to ensure good stability of the reduced order model. We show the importance of this a priori enrichment of the reduced basis, on a typical aeronautical injector with Reynolds number of 45,000. We demonstrate the capacity of this order reduction technique to recover large scale features for very long integration times (25 ms in our case). Moreover, the reduced order modeling (ROM) exhibits periodic fluctuations with a period of

2.2

ms corresponding to the time scale of the precessing vortex core (PVC) associated with this test case. We will end this paper by giving some prospects on the use of this stable reduced model in order to perform time extrapolation, that could be a strategy to study the limit cycle of the PVC. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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17 pages, 564 KiB

Open AccessArticle

Finite Symmetries in Agent-Based Epidemic Models

by Gilberto M. Nakamura, Ana Carolina P. Monteiro, George C. Cardoso and Alexandre S. Martinez

Math. Comput. Appl. 2019, 24(2), 44; https://doi.org/10.3390/mca24020044 - 23 Apr 2019

Cited by 4 | Viewed by 3203

Abstract

Predictive analysis of epidemics often depends on the initial conditions of the outbreak, the structure of the afflicted population, and population size. However, disease outbreaks are subjected to fluctuations that may shape the spreading process. Agent-based epidemic models mitigate the issue by using [...] Read more.

Predictive analysis of epidemics often depends on the initial conditions of the outbreak, the structure of the afflicted population, and population size. However, disease outbreaks are subjected to fluctuations that may shape the spreading process. Agent-based epidemic models mitigate the issue by using a transition matrix which replicates stochastic effects observed in real epidemics. They have met considerable numerical success to simulate small scale epidemics. The problem grows exponentially with population size, reducing the usability of agent-based models for large scale epidemics. Here, we present an algorithm that explores permutation symmetries to enhance the computational performance of agent-based epidemic models. Our findings bound the stochastic process to a single eigenvalue sector, scaling down the dimension of the transition matrix to

o (N^{2})

. Full article

(This article belongs to the Special Issue Dynamics Days Latin America and the Caribbean 2018)

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26 pages, 691 KiB

Open AccessFeature PaperArticle

Symplectic Model Order Reduction with Non-Orthonormal Bases

by Patrick Buchfink, Ashish Bhatt and Bernard Haasdonk

Math. Comput. Appl. 2019, 24(2), 43; https://doi.org/10.3390/mca24020043 - 21 Apr 2019

Cited by 24 | Viewed by 4226

Abstract

Parametric high-fidelity simulations are of interest for a wide range of applications. However, the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, [...] Read more.

Parametric high-fidelity simulations are of interest for a wide range of applications. However, the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g., structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such an ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As a new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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16 pages, 6045 KiB

Open AccessArticle

Investigation of Details in the Transition to Synchronization in Complex Networks by Using Recurrence Analysis

by Roberto C. Budzinski, Bruno Rafael Reichert Boaretto, Thiago L. Prado and Sergio Roberto Lopes

Math. Comput. Appl. 2019, 24(2), 42; https://doi.org/10.3390/mca24020042 - 20 Apr 2019

Cited by 2 | Viewed by 2983

Abstract

The study of synchronization in complex networks is useful for understanding a variety of systems, including neural systems. However, the properties of the transition to synchronization are still not well known. In this work, we analyze the details of the transition to synchronization [...] Read more.

The study of synchronization in complex networks is useful for understanding a variety of systems, including neural systems. However, the properties of the transition to synchronization are still not well known. In this work, we analyze the details of the transition to synchronization in complex networks composed of bursting oscillators under small-world and scale-free topologies using recurrence quantification analysis, specifically the determinism. We demonstrate the existence of non-stationarity states in the transition region. In the small-world network, the transition region denounces the existence of two-state intermittency. Full article

(This article belongs to the Special Issue Dynamics Days Latin America and the Caribbean 2018)

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28 pages, 6635 KiB

Open AccessFeature PaperArticle

An Error Indicator-Based Adaptive Reduced Order Model for Nonlinear Structural Mechanics—Application to High-Pressure Turbine Blades

by Fabien Casenave and Nissrine Akkari

Math. Comput. Appl. 2019, 24(2), 41; https://doi.org/10.3390/mca24020041 - 19 Apr 2019

Cited by 5 | Viewed by 3600

Abstract

The industrial application motivating this work is the fatigue computation of aircraft engines’ high-pressure turbine blades. The material model involves nonlinear elastoviscoplastic behavior laws, for which the parameters depend on the temperature. For this application, the temperature loading is not accurately known and [...] Read more.

The industrial application motivating this work is the fatigue computation of aircraft engines’ high-pressure turbine blades. The material model involves nonlinear elastoviscoplastic behavior laws, for which the parameters depend on the temperature. For this application, the temperature loading is not accurately known and can reach values relatively close to the creep temperature: important nonlinear effects occur and the solution strongly depends on the used thermal loading. We consider a nonlinear reduced order model able to compute, in the exploitation phase, the behavior of the blade for a new temperature field loading. The sensitivity of the solution to the temperature makes the classical unenriched proper orthogonal decomposition method fail. In this work, we propose a new error indicator, quantifying the error made by the reduced order model in computational complexity independent of the size of the high-fidelity reference model. In our framework, when the error indicator becomes larger than a given tolerance, the reduced order model is updated using one time step solution of the high-fidelity reference model. The approach is illustrated on a series of academic test cases and applied on a setting of industrial complexity involving five million degrees of freedom, where the whole procedure is computed in parallel with distributed memory. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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28 pages, 3279 KiB

Open AccessFeature PaperArticle

An Artificial Neural Network Based Solution Scheme for Periodic Computational Homogenization of Electrostatic Problems

by Felix Selim Göküzüm, Lu Trong Khiem Nguyen and Marc-André Keip

Math. Comput. Appl. 2019, 24(2), 40; https://doi.org/10.3390/mca24020040 - 17 Apr 2019

Cited by 8 | Viewed by 4971

Abstract

The present work addresses a solution algorithm for homogenization problems based on an artificial neural network (ANN) discretization. The core idea is the construction of trial functions through ANNs that fulfill a priori the periodic boundary conditions of the microscopic problem. A global [...] Read more.

The present work addresses a solution algorithm for homogenization problems based on an artificial neural network (ANN) discretization. The core idea is the construction of trial functions through ANNs that fulfill a priori the periodic boundary conditions of the microscopic problem. A global potential serves as an objective function, which by construction of the trial function can be optimized without constraints. The aim of the new approach is to reduce the number of unknowns as ANNs are able to fit complicated functions with a relatively small number of internal parameters. We investigate the viability of the scheme on the basis of one-, two- and three-dimensional microstructure problems. Further, global and piecewise-defined approaches for constructing the trial function are discussed and compared to finite element (FE) and fast Fourier transform (FFT) based simulations. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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17 pages, 342 KiB

Open AccessArticle

On Determination of Wave Velocities through the Eigenvalues of Material Objects

by Mikhail U. Nikabadze, Sergey A. Lurie, Hovik A. Matevossian and Armine R. Ulukhanyan

Math. Comput. Appl. 2019, 24(2), 39; https://doi.org/10.3390/mca24020039 - 11 Apr 2019

Cited by 2 | Viewed by 2563

Abstract

The statement of the eigenvalue problem for a tensor–block matrix of any order and of any
even rank is formulated. It is known that the eigenvalues of the tensor and the tensor–block matrix
are invariant quantities. Therefore, in this work, our goal is [...] Read more.

The statement of the eigenvalue problem for a tensor–block matrix of any order and of any
even rank is formulated. It is known that the eigenvalues of the tensor and the tensor–block matrix
are invariant quantities. Therefore, in this work, our goal is to find the expression for the velocities of
wave propagation of some medias through the eigenvalues of the material objects. In particular, we
consider the classical and micropolar materials with the different anisotropy symbols and for them
we determine the expressions for the velocities of wave propagation through the eigenvalues of the
material objects. Full article

(This article belongs to the Special Issue Mathematical Modeling in Physical Sciences)

13 pages, 1467 KiB

Open AccessArticle

Nonlocal FEM Formulation for Vibration Analysis of Nanowires on Elastic Matrix with Different Materials

by Büşra Uzun and Ömer Civalek

Math. Comput. Appl. 2019, 24(2), 38; https://doi.org/10.3390/mca24020038 - 6 Apr 2019

Cited by 30 | Viewed by 3319

Abstract

In this study, free vibration behaviors of various embedded nanowires made of different materials are investigated by using Eringen’s nonlocal elasticity theory. Silicon carbide nanowire (SiCNW), silver nanowire (AgNW), and gold nanowire (AuNW) are modeled as Euler–Bernoulli nanobeams with various boundary conditions such [...] Read more.

In this study, free vibration behaviors of various embedded nanowires made of different materials are investigated by using Eringen’s nonlocal elasticity theory. Silicon carbide nanowire (SiCNW), silver nanowire (AgNW), and gold nanowire (AuNW) are modeled as Euler–Bernoulli nanobeams with various boundary conditions such as simply supported (S-S), clamped simply supported (C-S), clamped–clamped (C-C), and clamped-free (C-F). The interactions between nanowires and medium are simulated by the Winkler elastic foundation model. The Galerkin weighted residual method is applied to the governing equations to gain stiffness and mass matrices. The results are given by tables and graphs. The effects of small-scale parameters, boundary conditions, and foundation parameters on frequencies are examined in detail. In addition, the influence of temperature change on the vibrational responses of the nanowires are also pursued as a case study. Full article

(This article belongs to the Special Issue Mesh-Free and Finite Element-Based Methods for Structural Mechanics Applications)

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13 pages, 3196 KiB

Open AccessArticle

Observable for a Large System of Globally Coupled Excitable Units

by Santiago Boari, Gonzalo Uribarri, Ana Amador and Gabriel B. Mindlin

Math. Comput. Appl. 2019, 24(2), 37; https://doi.org/10.3390/mca24020037 - 6 Apr 2019

Cited by 3 | Viewed by 3451

Abstract

The study of large arrays of coupled excitable systems has largely benefited from a technique proposed by Ott and Antonsen, which results in a low dimensional system of equations for the system’s order parameter. In this work, we show how to explicitly introduce [...] Read more.

The study of large arrays of coupled excitable systems has largely benefited from a technique proposed by Ott and Antonsen, which results in a low dimensional system of equations for the system’s order parameter. In this work, we show how to explicitly introduce a variable describing the global synaptic activation of the network into these family of models. This global variable is built by adding realistic synaptic time traces. We propose that this variable can, under certain conditions, be a good proxy for the local field potential of the network. We report experimental, in vivo, electrophysiology data supporting this claim. Full article

(This article belongs to the Special Issue Dynamics Days Latin America and the Caribbean 2018)

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Math. Comput. Appl., Volume 24, Issue 2 (June 2019) – 33 articles

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