ISWFoam: a numerical model for internal solitary wave simulation in continuously stratified fluids

A numerical model, ISWFoam, for simulating internal solitary waves (ISWs) in continuously stratified, incompressible, viscous fluids is developed based on a fully three-dimensional (3D) Navier–Stokes equation using the open-source code OpenFOAM®. This model combines the density transport equation with the Reynolds-averaged Navier–Stokes equation with the Coriolis force, and the model discrete equation adopts the finite-volume method. The k–ω SST turbulence model has also been modified according to the variable density field. ISWFoam provides two initial wave generation methods to generate an ISW in continuously stratified fluids, including solving the weakly nonlinear models of the extended Korteweg–de Vries (eKdV) equation and the fully nonlinear models of the Dubreil–Jacotin–Long (DJL) equation. Grid independence tests for ISWFoam are performed, and considering the accuracy and computing efficiency, the appropriate grid size of the ISW simulation is recommended to be 1/150th of the characteristic length and 1/25th of the ISW amplitude. Model verifications are conducted through comparisons between the simulated and experimental data for ISW propagation examples over a flat bottom section, including laboratory scale and actual ocean scale, a submerged triangular ridge, a Gaussian ridge, and slope. The laboratory test results, including the ISW profile, wave breaking location, ISW arrival time, and the spatial and temporal changes in the mixture region, are well reproduced by ISWFoam. The ISWFoam model with unstructured grids and local mesh refinement can effectively simulate the evolution of ISWs, the ISW breaking phenomenon, waveform inversion of ISWs, and the interaction between ISWs and complex topography.

A possible predictive mathematical model for the growth of a periphytic alga

Algae are photosynthetic organisms and have qualities that are very attractive for cultivation and industrial development for commercial purposes. When algal growth is analyzed for the production of biomass usually only the exponential phase of the growth curve is considered and the other phases are ignored. The objective of the work is to present a possible predictive mathematical model that allows a better understanding of the kinetic behavior of a periphytic microalgae by means of the use of the Smoluchowski discrete equation, with special emphasis on the lag phase. More specifically, unknown connection between the discrete Smoluchowski equation and the deterministic Baranyi model is shown in the present study. Analysis of this connection leads to a possible predictive mathematical model about of the kinetic behavior of a periphytic microalgae.

Reply to Martino’s comments on “The normal, the natural, and the harmonic”

This is my reply to Martino's comments on my publication “The normal, the natural, and the harmonic.” In this reply I show that the chaos equation, also known as the logistic discrete equation, is the same as the discretized logistic differential equation.

Case Study on the Fitting Method of Typical Objects

This study proposes different fitting methods for different types of targets in the 400–900 nm wavelength range, based on convex optimization algorithms, to achieve the effect of high-precision spectral reconstruction for small space-borne spectrometers. This article first expounds on the mathematical model in the imaging process of the small spectrometer and discretizes it into an AX = B matrix equation. Second, the design basis of the filter transmittance curve is explained. Furthermore, a convex optimization algorithm is used, based on 50 filters, and appropriate constraints are added to solve the target spectrum. First, in terms of spectrum fitting, six different ground object spectra are selected, and Gaussian fitting, polynomial fitting, and Fourier fitting are used to fit the original data and analyze the best fit of each target spectrum. Then the transmittance curve of the filter is equally divided, and the corresponding AX = B discrete equation set is obtained for the specific object target, and a random error of 1% is applied to the equation set to obtain the discrete spectral value. The fitting is performed for each case to determine the best fitting method with errors. Subsequently, the transmittance curve of the filter with the detector characteristics is equally divided, and the corresponding AX = B discrete equation set is obtained for the specific object target. A random error of 1% is applied to the equation set to obtain the error. After the discrete spectral values are obtained, the fitting is performed again, and the best fitting method is determined. In order to evaluate the fitting accuracy of the original spectral data and the reconstruction accuracy of the calculated discrete spectrum, the three evaluation indicators MSE, ARE, and RE are used for evaluation. To measure the stability and accuracy of the spectral reconstruction of the fitting method more accurately, it is necessary to perform 500 cycles of calculations to determine the corresponding MSE value and further analyze the influence of the fitting method on the reconstruction accuracy. The results show that different fitting methods should be adopted for different ground targets under the error conditions. The three indicators, MSE, ARE, and RE, have reached high accuracy and strong stability. The effect of high-precision reconstruction of the target spectrum is achieved. This article provides new ideas for related scholars engaged in hyperspectral reconstruction work and promotes the development of hyperspectral technology.

Numerical approximation of control problems of non-monotone and non-coercive semilinear elliptic equations

We analyze the numerical approximation of a control problem governed by a non-monotone and non-coercive semilinear elliptic equation. The lack of monotonicity and coercivity is due to the presence of a convection term. First, we study the finite element approximation of the partial differential equation. While we can prove existence of a solution for the discrete equation when the discretization parameter is small enough, the uniqueness is an open problem for us if the nonlinearity is not globally Lipschitz. Nevertheless, we prove the existence and uniqueness of a sequence of solutions bounded in $$L^\infty (\varOmega )$$ L ∞ ( Ω ) and converging to the solution of the continuous problem. Error estimates for these solutions are obtained. Next, we discretize the control problem. Existence of discrete optimal controls is proved, as well as their convergence to solutions of the continuous problem. The analysis of error estimates is quite involved due to the possible non-uniqueness of the discrete state for a given control. To overcome this difficulty we define an appropriate discrete control-to-state mapping in a neighbourhood of a strict solution of the continuous control problem. This allows us to introduce a reduced functional and obtain first order optimality conditions as well as error estimates. Some numerical experiments are included to illustrate the theoretical results.

ISWFoam: A numerical model for internal solitary wave simulation in continuously stratified fluids

A numerical model, ISWFoam, for simulating internal solitary waves (ISWs) in continuously stratified, incompressible, viscous fluids is developed based on a fully three-dimensional (3D) Navier-Stokes equation using the open source code OpenFOAM. This model combines the density transport equation with the Reynolds-averaged Navier-Stokes equation with the Coriolis force, and the model discrete equation adopts the finite volume method. The k-ω SST turbulence model has also been modified accordingly to the variable density field. ISWFoam provides two initial wave generation methods to generate an ISW in continuously stratified fluids, including solving the weakly nonlinear models of the extended Korteweg–de Vries (eKdV) equation and the fully nonlinear models of the Dubreil-Jacotin-Long (DJL) equation. Grid independence tests for ISWFoam are performed, considering the accuracy and computing efficiency, the appropriate grid size of the ISW simulation is recommended to be one-one hundred and fiftieth of the characteristic length and one-twenty fifth of the ISW amplitude. Model verifications are conducted through comparisons between the simulated and experimental data for ISW propagation examples over a flat bottom section, including laboratory scale and actual ocean scale, a submerged triangular ridge, a Gaussian ridge and slope. The laboratory test results, including the ISW profile, wave breaking location, ISW arrival time, and the spatial and temporal changes in the mixture region, are well reproduced by ISWFoam. The ISWFoam model with unstructured grids and local mesh refinement can accurately simulate the generation and evolution of ISWs, the ISW breaking phenomenon and the interaction between ISWs and complex structures and topography.

New Results on Qualitative Behavior of Second Order Nonlinear Neutral Impulsive Differential Systems with Canonical and Non-Canonical Conditions

The oscillation of impulsive differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of impulsive differential equations. In this work, several sufficient conditions are established for oscillatory or asymptotic behavior of second-order neutral impulsive differential systems for various ranges of the bounded neutral coefficient under the canonical and non-canonical conditions. Here, one can see that if the differential equations is oscillatory (or converges to zero asymptotically), then the discrete equation of similar type do not disturb the oscillatory or asymptotic behavior of the impulsive system, when impulse satisfies the discrete equation. Further, some illustrative examples showing applicability of the new results are included.

Big Data Tracking and Automatic Measurement Technology for Unmanned Aerial Vehicle Trajectory Based on MEMS Sensor

Due to the high cost and large error of traditional UAV big data tracking and automatic measurement technology, a method of big data tracking and automatic measurement for UAV trajectory based on MEMS sensor was put forward. The iterative learning control algorithm was used to estimate the repetitive disturbance and modeling error of system based on the simplified dynamics model of four-rotor helicopter and the optimal estimation characteristics of Kalman filter. The discrete equation of quadratic performance function in time domain was selected to compensate the estimated model error disturbance, and then the big data tracking was completed. Based on the data of gyroscope, the quaternion differential equation was established. The differential equation was solved by first-order Picard method, and a set of quaternion data was obtained. The gradient descent method was used to process the acceleration data and magnetic data, and thus to get the optimal quaternion. Finally, the measurement results were obtained by fusing the two quaternions with MEMS sensors. Simulation results prove that the proposed method can obtain the trajectory tracking data and measurement information of UAV accurately.

Bifurcation and chaos of axially moving nanobeams considering two scale effects based on non-local strain gradient theory

The nonlinear vibration of axially moving nanobeams at the microscale exhibits remarkable scale effects. A model of an axially moving nanobeam is established based on non-local strain gradient theory and considering two scale effects. The discrete equation of a non-autonomous planar system is obtained using the Galerkin method. The response characteristics of the system are determined using phase diagrams and Poincaré sections, and the effects of the scale parameters on the form of the motion are analyzed. The results show that as the non-local parameter and the material characteristic length parameter vary, the system undergoes multiple forms of motion, including periodic, period-doubling and chaotic motions. Two routes to chaos — period-doubling bifurcation and intermittent chaos — are identified in the variation ranges of the two scale parameters.