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# Dynamics, Volume 4, Issue 1 (March 2024) – 11 articles

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16 pages, 594 KiB
Article
Iterated Crank–Nicolson Method for Peridynamic Models
by Jinjie Liu, Samuel Appiah-Adjei and Moysey Brio
Dynamics 2024, 4(1), 192-207; https://doi.org/10.3390/dynamics4010011 - 15 Mar 2024
Viewed by 459
Abstract
In this paper, we explore the iterated Crank–Nicolson (ICN) algorithm for the one-dimensional peridynamic model. The peridynamic equation of motion is an integro-differential equation that governs structural deformations such as fractures. The ICN method was originally developed for hyperbolic advection equations. In peridynamics, [...] Read more.
In this paper, we explore the iterated Crank–Nicolson (ICN) algorithm for the one-dimensional peridynamic model. The peridynamic equation of motion is an integro-differential equation that governs structural deformations such as fractures. The ICN method was originally developed for hyperbolic advection equations. In peridynamics, we apply the ICN algorithm for temporal discretization and the midpoint quadrature method for spatial integration. Several numerical tests are carried out to evaluate the performance of the ICN method. In general, the ICN method demonstrates second-order accuracy, consistent with the Störmer–Verlet (SV) method. When the weight is 1/3, the ICN method behaves as a third-order Runge–Kutta method and maintains strong stability-preserving (SSP) properties for linear problems. Regarding energy conservation, the ICN algorithm maintains at least second-order accuracy, making it superior to the SV method, which converges linearly. Furthermore, selecting a weight of 0.25 results in fourth-order superconvergent energy variation for the ICN method. In this case, the ICN method exhibits energy variation similar to that of the fourth-order Runge–Kutta method but operates approximately 20% faster. Higher-order convergence for energy can also be achieved by increasing the number of iterations in the ICN method. Full article
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22 pages, 645 KiB
Article
A Novel Scalable Quantum Protocol for the Dining Cryptographers Problem
by Peristera Karananou and Theodore Andronikos
Dynamics 2024, 4(1), 170-191; https://doi.org/10.3390/dynamics4010010 - 8 Mar 2024
Viewed by 483
Abstract
This paper presents an innovative entanglement-based protocol to address the Dining Cryptographers problem, utilizing maximally entangled $|GH{Z}_{n}⟩$ tuples as its core. This protocol aims to provide scalability in terms of both the number of cryptographers n and the [...] Read more.
This paper presents an innovative entanglement-based protocol to address the Dining Cryptographers problem, utilizing maximally entangled $|GH{Z}_{n}⟩$ tuples as its core. This protocol aims to provide scalability in terms of both the number of cryptographers n and the amount of anonymous information conveyed, represented by the number of qubits m within each quantum register. The protocol supports an arbitrary number of cryptographers n, enabling scalability in both participant count and the volume of anonymous information transmitted. While the original Dining Cryptographers problem focused on a single bit of information—whether a cryptographer paid for dinner—the proposed protocol allows m, the number of qubits in each register, to be any arbitrarily large positive integer. This flexibility allows the transmission of additional information, such as the cost of the dinner or the timing of the arrangement. Another noteworthy aspect of the introduced protocol is its versatility in accommodating both localized and distributed versions of the Dining Cryptographers problem. The localized scenario involves all cryptographers gathering physically at the same location, such as a local restaurant, simultaneously. In contrast, the distributed scenario accommodates cryptographers situated in different places, engaging in a virtual dinner at the same time. Finally, in terms of implementation, the protocol accomplishes uniformity by requiring that all cryptographers utilize identical private quantum circuits. This design establishes a completely modular quantum system where all modules are identical. Furthermore, each private quantum circuit exclusively employs the widely used Hadamard and CNOT quantum gates, facilitating straightforward implementation on contemporary quantum computers. Full article
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13 pages, 2489 KiB
Article
Phase Diagram of Nuclear Pastas in Neutron Star Crusts
by Jorge A. Muñoz and Jorge A. López
Dynamics 2024, 4(1), 157-169; https://doi.org/10.3390/dynamics4010009 - 22 Feb 2024
Viewed by 660
Abstract
Two neural networks were trained to predict, respectively, the Euler characteristic and the curvature of nuclear pastas in neutron star crust conditions generated by molecular dynamics simulations of neutron star matter with 0.1 < x < 0.5, 0.040 fm−3 < ρ < [...] Read more.
Two neural networks were trained to predict, respectively, the Euler characteristic and the curvature of nuclear pastas in neutron star crust conditions generated by molecular dynamics simulations of neutron star matter with 0.1 < x < 0.5, 0.040 fm−3 < ρ < 0.085 fm−3 (0.68 × 1014 g/cm3 < ρ < 1.43 × 1014 g/cm3), and 0.2 MeV < T < 4.0 MeV, where x is proton content, the density is $\rho$, and the temperature is T. The predictions of the two networks were combined to determine the nuclear pasta phase that is thermodynamically stable at a given x, $\rho$, and T, and a three-dimensional phase diagram that extrapolated slightly the regions of existing molecular dynamics data was computed. The jungle gym and anti-jungle gym structures are prevalent at high temperature and low density, while the anti-jungle gym and anti-gnocchi structures dominate at high temperature and high density. A diversity of structures exist at low temperatures and intermediate density and proton content. The trained models used in this work are open access and available at a public repository to promote comparison to pastas obtained with other models. Full article
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22 pages, 10110 KiB
Article
Development and Validation of a Compressible Reacting Gas-Dynamic Flow Solver for Supersonic Combustion
by Anvar Gilmanov, Ponnuthurai Gokulakrishnan and Michael S. Klassen
Dynamics 2024, 4(1), 135-156; https://doi.org/10.3390/dynamics4010008 - 11 Feb 2024
Viewed by 889
Abstract
An approach based on the OpenFOAM library has been developed to solve a high-speed, multicomponent mixture of a reacting, compressible flow. This work presents comprehensive validation of the newly developed solver, called compressibleCentralReactingFoam, with different supersonic flows, including shocks, expansion waves, and [...] Read more.
An approach based on the OpenFOAM library has been developed to solve a high-speed, multicomponent mixture of a reacting, compressible flow. This work presents comprehensive validation of the newly developed solver, called compressibleCentralReactingFoam, with different supersonic flows, including shocks, expansion waves, and turbulence–combustion interaction. The comparisons of the simulation results with experimental and computational data confirm the fidelity of this solver for problems involving multicomponent high-speed reactive flows. The gas dynamics of turbulence–chemistry interaction are modeled using a partially stirred reactor formulation and provide promising results to better understand the complex physics involved in supersonic combustors. A time-scale analysis based on local Damköhler numbers reveals different regimes of turbulent combustion. In the core of the jet flow, the Damköhler number is relatively high, indicating that the reaction time scale is smaller than the turbulent mixing time scale. This means that the combustion is controlled by turbulent mixing. In the shear layer, where the heat release rate and the scalar dissipation rate have the highest value, the flame is stabilized due to finite rate chemistry with small Damköhler numbers and a limited fraction of fine structure. This solver allows three-dimensional gas dynamic simulation of high-speed multicomponent reactive flows relevant to practical combustion applications. Full article
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16 pages, 12243 KiB
Article
Physical Reservoir Computing Enabled by Solitary Waves and Biologically Inspired Nonlinear Transformation of Input Data
by Ivan S. Maksymov
Dynamics 2024, 4(1), 119-134; https://doi.org/10.3390/dynamics4010007 - 8 Feb 2024
Cited by 1 | Viewed by 789
Abstract
Reservoir computing (RC) systems can efficiently forecast chaotic time series using the nonlinear dynamical properties of an artificial neural network of random connections. The versatility of RC systems has motivated further research on both hardware counterparts of traditional RC algorithms and more-efficient RC-like [...] Read more.
Reservoir computing (RC) systems can efficiently forecast chaotic time series using the nonlinear dynamical properties of an artificial neural network of random connections. The versatility of RC systems has motivated further research on both hardware counterparts of traditional RC algorithms and more-efficient RC-like schemes. Inspired by the nonlinear processes in a living biological brain and using solitary waves excited on the surface of a flowing liquid film, in this paper, we experimentally validated a physical RC system that substitutes the effect of randomness that underpins the operation of the traditional RC algorithm for a nonlinear transformation of input data. Carrying out all operations using a microcontroller with minimal computational power, we demonstrate that the so-designed RC system serves as a technically simple hardware counterpart to the ‘next-generation’ improvement of the traditional RC algorithm. Full article
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22 pages, 12671 KiB
Article
Analysis of Wind Turbine Wake Dynamics by a Gaussian-Core Vortex Lattice Technique
by Apurva Baruah and Fernando Ponta
Dynamics 2024, 4(1), 97-118; https://doi.org/10.3390/dynamics4010006 - 1 Feb 2024
Cited by 3 | Viewed by 768
Abstract
The development and deployment of the next generation of wind energy systems calls for simulation tools that model the entire wind farm while balancing accuracy and computational cost. A full-system wind farm simulation must consider the atmospheric inflow, the wakes and consequent response [...] Read more.
The development and deployment of the next generation of wind energy systems calls for simulation tools that model the entire wind farm while balancing accuracy and computational cost. A full-system wind farm simulation must consider the atmospheric inflow, the wakes and consequent response of the multiple turbines, and the implementation of the appropriate farm-collective control strategies that optimize the entire wind farm’s output. In this article, we present a novel vortex lattice model that enables the effective representation of the complex vortex wake dynamics of the turbines in a farm subject to transient inflow conditions. This work extends the capabilities of our multi-physics suite, CODEF, to include the capability to simulate the wakes and the high-fidelity aeroelastic response of multiple turbines in a wind farm. Herein, we compare the results of our GVLM technique with the LiDAR measurements obtained at Sandia National Laboratories’ SWiFT facility. The comparison shows remarkable similarities between the simulation and field measurements of the wake velocity. These similarities demonstrate our model’s capabilities in capturing the entire wake of a wind turbine at a significantly reduced computational cost as compared to other techniques. Full article
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16 pages, 206 KiB
Article
New Types of Derivative Non-linear Schrödinger Equations Related to Kac–Moody Algebra $A2(1)$
by Aleksander Aleksiev Stefanov
Dynamics 2024, 4(1), 81-96; https://doi.org/10.3390/dynamics4010005 - 18 Jan 2024
Viewed by 429
Abstract
We derive a new system of integrable derivative non-linear Schrödinger equations with an L operator, quadratic in the spectral parameter with coefficients belonging to the Kac–Moody algebra ${A}_{2}^{\left(1\right)}$. The construction of the fundamental analytic solutions of L is [...] Read more.
We derive a new system of integrable derivative non-linear Schrödinger equations with an L operator, quadratic in the spectral parameter with coefficients belonging to the Kac–Moody algebra ${A}_{2}^{\left(1\right)}$. The construction of the fundamental analytic solutions of L is outlined and they are used to introduce the scattering data, thus formulating the scattering problem for the Lax pair $L,M$. Full article
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24 pages, 2351 KiB
Article
Non-Symmetry in the Shock Refraction at a Closed Interface as a Recovery Mechanism
by Anna Markhotok
Dynamics 2024, 4(1), 57-80; https://doi.org/10.3390/dynamics4010004 - 10 Jan 2024
Viewed by 702
Abstract
The possibility of a shock wave recovery at a discrete closed interface with a heated gas has been investigated. A two-dimensional model applied to conditions of optical discharges featuring spherical, elliptical, and drop-like configurations demonstrated that non-symmetry in the shock refraction contributes to [...] Read more.
The possibility of a shock wave recovery at a discrete closed interface with a heated gas has been investigated. A two-dimensional model applied to conditions of optical discharges featuring spherical, elliptical, and drop-like configurations demonstrated that non-symmetry in the shock refraction contributes to the specific mechanism of recovery other than simply its compensation. Even though the full restoration of the hypersonic flow state does not occur in a strict sense of it, clear reverse changes toward the initial shape of the shock front eventually take place, thus creating an appearance of a full recovery seen in experiments. From analysis of different interface symmetries, the factors determining the recovery dynamics are identified. The results are directly applicable to the problem of energy deposition into a hypersonic flow; however, it can be useful anywhere else where the flow modifications following the interaction are important. The dimensionless form of the equations allows applications on any scale other than that demonstrated for the optical discharges. Full article
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17 pages, 618 KiB
Article
Multiplicative Renormalization of Stochastic Differential Equations for the Abelian Sandpile Model
by Dimitri Volchenkov
Dynamics 2024, 4(1), 40-56; https://doi.org/10.3390/dynamics4010003 - 4 Jan 2024
Viewed by 969
Abstract
The long-term, large-scale behavior in a problem of stochastic nonlinear dynamics corresponding to the Abelian sandpile model is studied with the use of the quantum-field theory renormalization group approach. We prove the multiplicative renormalization of the model including an infinite number of coupling [...] Read more.
The long-term, large-scale behavior in a problem of stochastic nonlinear dynamics corresponding to the Abelian sandpile model is studied with the use of the quantum-field theory renormalization group approach. We prove the multiplicative renormalization of the model including an infinite number of coupling parameters, calculate an infinite number of renormalization constants, identify a plane of fixed points in the infinite dimensional space of coupling parameters, discuss their stability and critical scaling in the model, and formulate a simple law relating the asymptotic size of an avalanche to a model exponent quantifying the time-scale separation between the slow energy injection and fast avalanche relaxation processes. Full article
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26 pages, 11420 KiB
Article
Implementation and Validation of Explicit Immersed Boundary Method and Lattice Boltzmann Flux Solver in OpenFOAM
by Yangyang Liu, Ziying Zhang, Hua Zhang and Yaguang Liu
Dynamics 2024, 4(1), 14-39; https://doi.org/10.3390/dynamics4010002 - 3 Jan 2024
Viewed by 813
Abstract
In this work, the explicit boundary-condition-enforced immersed boundary method (EIBM) and the lattice Boltzmann flux solver (LBFS) are integrated into OpenFOAM to efficiently solve incompressible flows with complex geometries and moving boundaries. The EIBM applies the explicit technique to greatly improve the computational [...] Read more.
In this work, the explicit boundary-condition-enforced immersed boundary method (EIBM) and the lattice Boltzmann flux solver (LBFS) are integrated into OpenFOAM to efficiently solve incompressible flows with complex geometries and moving boundaries. The EIBM applies the explicit technique to greatly improve the computational efficiency of the original boundary-condition-enforced immersed boundary method. In addition, the improved EIBM inherits the accurate interpretation of the no-slip boundary condition and the simple implementation from the original one. The LBFS uses the finite volume method to discretize the recovered macroscopic governing equations from the lattice Boltzmann equation. It enjoys the explicit relationship between the pressure and density, which avoids solving the pressure Poisson equation and thus saves much computational cost. Another attractive feature of the LBFS lies in its simultaneous evaluation of the inviscid and viscous fluxes. OpenFOAM, as an open-source CFD platform, has drawn increasing attention from the CFD community and has been proven to be a powerful tool for various problems. Thus, implementing the EIBM and LBFS into such a popular platform can advance the practical application of these two methods and may provide an effective alternative for complicated incompressible flow problems. The performance of the integrated solver in OpenFOAM is comprehensively assessed by comparing it with the widely used numerical solver in OpenFOAM, namely, the Pressure-Implicit with Splitting of Operators (PISO) algorithm with the IBM. A series of representative test cases with stationary and moving boundaries are simulated. Numerical results confirm that the present method does not have any streamline penetration and achieves the second-order accuracy in space. Therefore, the present method implemented in the open-source platform OpenFOAM may have good potential and can serve as a powerful tool for practical engineering problems. Full article
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13 pages, 1588 KiB
Article
Dynamics of Fricke–Painlevé VI Surfaces
by Michel Planat, David Chester and Klee Irwin
Dynamics 2024, 4(1), 1-13; https://doi.org/10.3390/dynamics4010001 - 2 Jan 2024
Cited by 1 | Viewed by 1105
Abstract
The symmetries of a Riemann surface $\mathsf{\Sigma }\setminus \left\{{a}_{i}\right\}$ with n punctures ${a}_{i}$ are encoded in its fundamental group ${\pi }_{1}\left(\mathsf{\Sigma }\right)$. Further structure may be described through representations (homomorphisms) of ${\pi }_{1}$ over a Lie [...] Read more.
The symmetries of a Riemann surface $\mathsf{\Sigma }\setminus \left\{{a}_{i}\right\}$ with n punctures ${a}_{i}$ are encoded in its fundamental group ${\pi }_{1}\left(\mathsf{\Sigma }\right)$. Further structure may be described through representations (homomorphisms) of ${\pi }_{1}$ over a Lie group G as globalized by the character variety $\mathcal{C}=\mathrm{Hom}\left({\pi }_{1},G\right)/G$. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the four-punctured Riemann sphere $\mathsf{\Sigma }={S}_{2}^{\left(4\right)}$ and the ‘space-time-spin’ group $G=S{L}_{2}\left(\mathbb{C}\right)$. In such a situation, $\mathcal{C}$ possesses remarkable properties: (i) a representation is described by a three-dimensional cubic surface ${V}_{a,b,c,d}\left(x,y,z\right)$ with three variables and four parameters; (ii) the automorphisms of the surface satisfy the dynamical (non-linear and transcendental) Painlevé VI equation (or ${P}_{VI}$); and (iii) there exists a finite set of 1 (Cayley–Picard)+3 (continuous platonic)+45 (icosahedral) solutions of ${P}_{VI}$. In this paper, we feature the parametric representation of some solutions of ${P}_{VI}$: (a) solutions corresponding to algebraic surfaces such as the Klein quartic and (b) icosahedral solutions. Applications to the character variety of finitely generated groups ${f}_{p}$ encountered in TQC or DNA/RNA sequences are proposed. Full article
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