In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and ...control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed.
In this study, the harmonic and 1/3 subharmonic oscillations of a single degree of freedom Duffing oscillator with large nonlinearity and large damping are investigated by using a simple point ...collocation method applied in the time domain over a period of the periodic solution. The relationship between the proposed collocation method and the high dimensional harmonic balance method (HDHB), proposed earlier by Thomas, Dowell, and Hall (2002), is explored. We demonstrate that the HDHB is not a kind of "harmonic balance method" but essentially a cumbersome version of the collocation method. In using the collocation method, the collocation-resulting nonlinear algebraic equations (NAEs) are solved by the Newton-Raphson method. To start the Newton iterative process, initial values for the N harmonics approximation are provided by solving the corresponding low order harmonic approximation with the aid of Mathematica. We also introduce a generating frequency (ωg), where by the response curves are effectively obtained. Amplitude-frequency response curves for various values of damping, nonlinearity, and force amplitude are obtained and compared to show the effect of each parameter. In addition, the time Galerkin method the Harmonic-Balance method is applied and compared with the presently proposed collocation method. Numerical examples confirm the simplicity and effectiveness of the present collocation scheme in the time domain.
Spin projected wave functions are generalizations of the Hartree–Fock wave function. Among them, the Half‐Projected Hartree–Fock (HPHF) wave function is a nearly pure wave function of spin and ...recovers a small part of the spin correlation energy. This paper reviews the history of the HPHF theory, not only from the conceptual point of view but also providing a compilation of the publications of this method over the years until now. In addition, the extension of the HPHF method to the calculation of excited states with the same symmetry as the ground state will be discussed. The possible variational collapse during the calculation of a singlet excited state of the same symmetry as the ground state is avoided by orthogonalizing a pair of corresponding orbitals, one occupied and one virtual orbital, at every step of the variational process. As an example, the potential energy surfaces of the S0 ground and 1S1(n, π*) first excited state of the formic acid HCOOH are calculated.
Derived from the Projected HF method, the Half‐Projected HF (HPHF) method is historically reviewed here. This wave function consists in two Slater determinants in which half of the spin contaminants have been removed. The wave function for a singlet state does not contain triplet spin contamination and vice versa. Possessing the simplicity of the RHF method and using the computational time of a UHF calculation, the HPHF method finds its maximum application in the calculation of excited states with the same symmetry as the ground state.
H ∞ control is a powerful method to solve the disturbance attenuation problems that occur in some control systems. The design of such controllers relies on solving the zero-sum game (ZSG). But in ...practical applications, the exact dynamics is mostly unknown. Identification of dynamics also produces errors that are detrimental to the control performance. To overcome this problem, an iterative adaptive dynamic programming algorithm is proposed in this paper to solve the continuous-time, unknown nonlinear ZSG with only online data. A model-free approach to the Hamilton-Jacobi-Isaacs equation is developed based on the policy iteration method. Control and disturbance policies and value are approximated by neural networks (NNs) under the critic-actor-disturber structure. The NN weights are solved by the least-squares method. According to the theoretical analysis, our algorithm is equivalent to a Gauss-Newton method solving an optimization problem, and it converges uniformly to the optimal solution. The online data can also be used repeatedly, which is highly efficient. Simulation results demonstrate its feasibility to solve the unknown nonlinear ZSG. When compared with other algorithms, it saves a significant amount of online measurement time.
•Analytical solutions of the integrable generalized (2+1)-dimensional nonlinear conformable Schrodinger system of equations.•The use of (G′/G)-expansion method, generalized Riccati equation mapping ...method and the Kudryashov method in the conformable sense.•To discover a new and more general variety of exact traveling wave solutions.•Several plots illustrating the behavior of dynamic shapes of the solutions.
The analytical solutions of the integrable generalized (2+1)-dimensional nonlinear conformable Schrödinger (NLCS) system of equations was explored in this paper with the aid of three novel techniques which consist of (G′/G)-expansion method, generalized Riccati equation mapping method and the Kudryashov method in the conformable sense. We have discovered a new and more general variety of exact traveling wave solutions by using the proposed methods with a variety of soliton solutions of several structures. With several plots illustrating the behavior of dynamic shapes of the solutions, the findings are highly applicable and detailed the physical dynamic of the considered nonlinear system.
The finite-element model is established to simulate electromagnetic forces of a permanent-magnet (PM) spherical motor using the commercial software from Ansoft. The Taguchi method is used to optimize ...the magnetic field density in the radial air gap and the maximum torque of the PM spherical motor. The final optimization parameters are determined by a response surface method (RSM). The simulation results of the optimized motor are compared with those of the original motor to show that the output performances of the optimized motor have improved greatly, which verifies the effectiveness and practicability of the Taguchi method on the optimization of the PM spherical motor.
This paper presents Hamiltonian finite element methods for approximating semilinear wave propagation problems, including the nonlinear Klein–Gordon and sine-Gordon equations. The aim is to obtain ...accurate high-order approximations while conserving physical quantities of interest such as energy. To achieve conservation properties at a discrete level, we propose semidiscrete schemes based on two Hamiltonian structures of the equation. These include Mixed finite element methods, discontinuous Galerkin methods, and hybridizable discontinuous Galerkin methods (HDG). In particular, we propose a new class of DG methods using time operators to define the numerical traces, ultimately leading to an energy-conserving scheme. Time discretization uses Symplectic explicit-partitioned and diagonally-implicit Runge–Kutta schemes. Furthermore, the paper showcases several numerical examples that demonstrate the accuracy and energy conservation properties of the approximations, along with the simulation of soliton cloning.
The performance of three machine learning methods (support vector regression, random forests and artificial neural network) for estimating the LAI of paddy rice was evaluated in this study. ...Traditional univariate regression models involving narrowband NDVI with optimized band combinations as well as linear multivariate calibration partial least squares regression models were also evaluated for comparison. A four year field-collected dataset was used to test the robustness of LAI estimation models against temporal variation. The partial least squares regression and three machine learning methods were built on the raw hyperspectral reflectance and the first derivative separately. Two different rules were used to determine the models' key parameters. The results showed that the combination of the red edge and NIR bands (766 nm and 830 nm) as well as the combination of SWIR bands (1114 nm and 1190 nm) were optimal for producing the narrowband NDVI. The models built on the first derivative spectra yielded more accurate results than the corresponding models built on the raw spectra. Properly selected model parameters resulted in comparable accuracy and robustness with the empirical optimal parameter and significantly reduced the model complexity. The machine learning methods were more accurate and robust than the VI methods and partial least squares regression. When validating the calibrated models against the standalone validation dataset, the VI method yielded a validation RMSE value of 1.17 for NDVI(766,830) and 1.01 for NDVI(1114,1190), while the best models for the partial least squares, support vector machine and artificial neural network methods yielded validation RMSE values of 0.84, 0.82, 0.67 and 0.84, respectively. The RF models built on the first derivative spectra with mtry = 10 showed the highest potential for estimating the LAI of paddy rice.
An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap (DSG) technique using triangular meshes (ES-DSG) was recently proposed to enhance the accuracy of the existing ...FEM with the DSG for analysis of isotropic Reissner/Mindlin plates. In this paper, the ES-DSG is further formulated for static, free vibration and buckling analyses of functionally graded material (FGM) plates. The thermal and mechanical properties of FGM plates are assumed to vary across the thickness of the plate by a simple power rule of the volume fractions of the constituents. In the ES-DSG, the stiffness matrices are obtained by using the strain smoothing technique over the smoothing domains associated with the edges of the elements. The present formulation uses only linear approximations and its implementation into finite element programs is quite simple. Several numerical examples are given to demonstrate the performance of the present formulation for FGM plates.
We develop an immersed meshfree method under a variational multiscale framework for modeling fluid–structure interactive systems involving shock waves. The proposed method enables flexible ...non-body-fitted discretization, approximations, and quadrature rules for solid and fluid subdomains. The interfacial compatibility conditions are imposed by a volumetric constraint, which avoids the tedious contour integral and interface tracking. The reproducing kernel particle method (RKPM) is employed for both solid and fluid sub-systems, which allows arbitrary control of the orders of continuity and approximation, as well as flexibility in discretization, making it particularly advantageous for modeling fluid–structure interaction (FSI). In the proposed approach, the fictitious fluid is combined with the foreground solid, forming an “effective solid problem” solved on a moving foreground domain, while the background fluid problem is solved with prescribed solid velocity in the overlapping domain to reduce the leaking instability and mesh sensitivity. The variational multiscale immersed method (VMIM) is employed to enhance accuracy and stability in FS coupling, which leads to a residual-based stabilization. The MUSCL-SCNI shock algorithm provides a natural way of introducing the Riemann solution in the shock algorithm via the SCNI contour integral for desirable accuracy. The employment of SCNI in the proposed framework also provides computational efficiency, accuracy, and stability. Using a larger RKPM support size in the fluid domain can effectively suppress the leaking instability. The effectiveness of the proposed methods is verified in solving several FSI problems with shock waves, and the enhanced stability and accuracy of the proposed methods compared to the classical immersed approach have also been demonstrated.
•A variational multiscale immersed method (VMIM) is proposed for FSI with shocks.•The improved immersed FSI formulation avoids the leaking instability.•The VMIM results in an immersed residual-based stabilization for FSI-coupling.•The RKPM is advantageous in constructing the high-order derivatives in VMIM.•The VMIM with MUSCL-SCNI enhances stability and accuracy in shock-capturing.
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