Moment Equations Research Articles

Identification of moment equations via data-driven approaches in nonlinear Schrödinger models

IntroductionThe moment quantities associated with the nonlinear Schrödinger equation offer important insights into the evolution dynamics of such dispersive wave partial differential equation (PDE) models. The effective dynamics of the moment quantities are amenable to both analytical and numerical treatments.MethodsIn this paper, we present a data-driven approach associated with the “Sparse Identification of Nonlinear Dynamics” (SINDy) to capture the evolution behaviors of such moment quantities numerically.Results and DiscussionOur method is applied first to some well-known closed systems of ordinary differential equations (ODEs) which describe the evolution dynamics of relevant moment quantities. Our examples are, progressively, of increasing complexity and our findings explore different choices within the SINDy library. We also consider the potential discovery of coordinate transformations that lead to moment system closure. Finally, we extend considerations to settings where a closed analytical form of the moment dynamics is not available.

Iterative Methods of Linearized Moment Equations for Rarefied Gases

Iterative Methods of Linearized Moment Equations for Rarefied Gases

DG-IMEX method for a two-moment model for radiation transport in the [formula omitted] limit

We consider neutral particle systems described by moments of a phase-space density and propose a realizability-preserving numerical method to evolve a spectral two-moment model for particles interacting with a background fluid moving with nonrelativistic velocities. The system of nonlinear moment equations, with special relativistic corrections to O(v/c), expresses a balance between phase-space advection and collisions and includes velocity-dependent terms that account for spatial advection, Doppler shift, and angular aberration. The model is conservative for the correct O(v/c) Eulerian-frame number density and is consistent, to O(v/c), with Eulerian-frame energy and momentum conservation. This model is closely related to the one promoted by Lowrie et al. [1] and similar to models currently used to study transport phenomena in large-scale simulations of astrophysical environments. The proposed numerical method is designed to preserve moment realizability, which guarantees that the moments correspond to a nonnegative phase-space density. The realizability-preserving scheme consists of the following key components: (i) a strong stability-preserving implicit-explicit (IMEX) time-integration method; (ii) a discontinuous Galerkin (DG) phase-space discretization with carefully constructed numerical fluxes; (iii) a realizability-preserving implicit collision update; and (iv) a realizability-enforcing limiter. In time integration, nonlinearity of the moment model necessitates solution of nonlinear equations, which we formulate as fixed-point problems and solve with tailored iterative solvers that preserve moment realizability with guaranteed global convergence. We also analyze the simultaneous Eulerian-frame number and energy conservation properties of the semi-discrete DG scheme and propose a “spectral redistribution” scheme that promotes Eulerian-frame energy conservation. Through numerical experiments, we demonstrate the accuracy and robustness of this DG-IMEX method and investigate its Eulerian-frame energy conservation properties.

Maximum likelihood estimation for left-truncated log-logistic distributions with a given truncation point

AbstractFor a sample $$X_1, X_2,\ldots X_N$$ X 1 , X 2 , … X N of independent identically distributed copies of a log-logistically distributed random variable X the maximum likelihood estimation is analysed in detail if a left-truncation point $$x_L>0$$ x L > 0 is introduced. Due to scaling properties it is sufficient to investigate the case $$x_L=1$$ x L = 1 . Here the corresponding maximum likelihood equations for a normalised sample (i.e. a sample divided by $$x_L$$ x L ) do not always possess a solution. A simple criterion guarantees the existence of a solution: Let $$\mathbb {E}(\cdot )$$ E ( · ) denote the expectation induced by the normalised sample and denote by $$\beta _0=\mathbb {E}(\ln {X})^{-1}$$ β 0 = E ( ln X ) - 1 , the inverse value of expectation of the logarithm of the sampled random variable X (which is greater than $$x_L=1$$ x L = 1 ). If this value $$\beta _0$$ β 0 is bigger than a certain positive number $$\beta _C$$ β C then a solution of the maximum likelihood equation exists. Here the number $$\beta _C$$ β C is the unique solution of a moment equation,$$\mathbb {E}(X^{-\beta _C})=\frac{1}{2}$$ E ( X - β C ) = 1 2 . In the case of existence a profile likelihood function can be constructed and the optimisation problem is reduced to one dimension leading to a robust numerical algorithm. When the maximum likelihood equations do not admit a solution for certain data samples, it is shown that the Pareto distribution is the $$L^1$$ L 1 -limit of the degenerated left-truncated log-logistic distribution, where $$L^1(\mathbb {R}^+)$$ L 1 ( R + ) is the usual Banach space of functions whose absolute value is Lebesgue-integrable. A large sample analysis showing consistency and asymptotic normality complements our analysis. Finally, two applications to real world data are presented.

A New Method for Solving Three Moment Equation in Well Trajectory Control

A New Method for Solving Three Moment Equation in Well Trajectory Control

Transient dispersion of settling gyrotactic microorganisms in an open channel flow

Dispersion of microorganisms is a key issue in bio-physics and has many applications in the fields of algae cultivation, biomass energy, and wetland ecology. However, there has been limited exploration of the effects of settling behavior and initial release conditions on the transient dispersion of gyrotactic microorganisms. This paper explores the transient dispersion of settling gyrotactic microorganisms in an open channel flow. The moment equations derived from the Smoluchowski equation are solved by the biorthogonal expansion method, and the results are compared with random walk simulations, showing good agreement. The time variations of concentration distribution, drift velocity, and dispersivity of settling gyrotactic microorganism suspension are explored in detail under typical initial release conditions. As illustrated and characterized, settlement weakens the gravitactic focusing of microorganisms near the free surface, leads to accumulation at the bottom, and increases the dispersivity; from a line source release, the relaxation time is shortest, and the microorganisms scatter fastest in the longitudinal direction, while the point source at the water surface leads to the most concentrated longitudinal distribution and the highest drift velocity; furthermore, the initial release condition assumes an important role in shaping the concentration distribution and drift velocity.

Stability, Uniqueness and Existence of Solutions to McKean–Vlasov Stochastic Differential Equations in Arbitrary Moments

We deduce stability and pathwise uniqueness for a McKean–Vlasov equation with random coefficients and a multidimensional Brownian motion as driver. Our analysis focuses on a non-Lipschitz continuous drift and includes moment estimates for random Itô processes that are of independent interest. For deterministic coefficients, we provide unique strong solutions even if the drift fails to be of affine growth. The theory that we develop rests on Itô’s formula and leads to pth moment and pathwise exponential stability for p≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 2$$\\end{document} with explicit Lyapunov exponents.

Bending of bidirectional functionally graded nonlocal stress-driven beam

This paper provides a comprehensive analysis, using nonlocal stress-driven integral theory, of the static behavior of a nanoscale beam of bidirectionally graded materials. After a brief explanation of the mathematical formulation of BDFGMs, the work done and strain energy expressions derived from the displacement field are discussed. Variational formulations and Hamilton's principle are used to develop the equilibrium equation. An analytical development of the nonlocal kernel for stress-driven integral theory and formulated governing equation which was nondimensionalized later. Explicit equations for displacement and moment are obtained by solving this equation using the Laplace transformation. Three different boundary conditions are examined, and differences in the maximum displacement with respect to the nonlocal parameter and the two material FGM parameters are displayed both visually and in table form. The results exhibit excellent agreement and provide a standard for further research when they are closely compared to the existing numerical data. This work contributes to the knowledge of BDFGMs under nonlocal effects generated by stress-driven integral theory and offers solutions that have been confirmed for further investigation.

Analysis of pipe fitting sealing torque with SolidWorks simulations

The tightness and stability of the pipe connections are particularly important for pressurized environments. Pipe fittings are one of the most important subgroups of fittings. These fittings are not only used for connection purposes, but these fasteners may also have additional purposes such as sealing. In this study, a simulation model of the pipe fitting was created to evaluate the sealing performance. The obtained forces in the simulations were then used in the modified version of the classical bolt-tightening moment equation. In this equation, the additional parameter, R, is embedded. This parameter depends on the distance from the center (torque arm, x), slope parameter (k), and friction constant (S) between the body or nut and the ring parts. The consistency and applicability of the new equation were discussed by comparing the obtained results with the experimental results. It has been observed that there is a good agreement between the comparative results.

Inverse Modeling of Heterogeneous Structures in Electron Probe Microanalysis.

Electron probe microanalysis (EPMA) is a powerful tool for chemical characterization of materials on a microscopic scale. However, EPMA has the drawback that its information volume has a spatial extent of some 100 nm to a few µm. With the introduction of new electron sources, i.e., Schottky Thermal Field and Cold Field Emitter, where the electron beam is focused down to a few nm, measurements can be nowadays performed on the sub-micrometer scale. The goal of the work is to reveal the chemical composition of structures smaller than the excitation volume. New strategies are presented where the acquisition is performed at different positions on the sample and as a scan across a fine structure by using one or more single beam energies. Besides the well-known Monte-Carlo simulation, a deterministic model is also used. The deterministic model is based on moment equations of the Boltzmann equation. Inverse modeling is presented for several case studies. Due to the highly complex nonlinearity of the inverse model, an ill-posed and well-posed problem is shown as well. Finally, the method is extended to reconstruct 2D structures, i.e., rectangular shaped particles, with heterogeneous composition on lateral and depth scale.

A Novel ES-BGK Model for Non-polytropic Gases with Internal State Density Independent of the Temperature

A novel ES-BGK-based model of non-polytropic rarefied gases in the framework of kinetic theory is presented. Key features of this model are: an internal state density function depending only on the microscopic energy of internal modes (avoiding the dependence on temperature seen in previous reference studies); full compliance with the H-theorem; feasibility of the closure of the system of moment equations based on the maximum entropy principle, following the well-established procedure of rational extended thermodynamics. The structure of planar shock waves in carbon dioxide (CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document}) obtained with the present model is in general good agreement with that of previous results, except for the computed internal temperature profile, which is qualitatively different with respect to the results obtained in previous studies, showing here a consistently monotonic behavior across the shock structure, rather than the non monotonic behavior previously found.

Probability density function of roll amplitude for parametric rolling using moment equation

For the ship stability estimation and ship design, it is helpful to know the probability of the roll amplitude exceeding a certain threshold. Therefore, it is necessary to obtain the probability density function of the roll amplitude. In this study, first, we derive the moment values of roll amplitude by combining the moment equations and the linearity of expectation. With this, we propose a method to estimate the probability density function of the roll amplitude by using the obtained moment values. The results of our proposed method are compared to those obtained from Monte Carlo simulations. When the higher-order cumulant neglect closure method is used, the moment values resulting from the moment equations are close to the results of corresponding Monte Carlo simulations. In addition, our proposed method for deriving the probability density function of the roll amplitude is validated by comparison with Monte Carlo simulation results. In conclusion, we can state that the proposed methods for deriving the moments and the probability density function of the roll amplitude are available for practical use cases.

Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: a study using POD-Galerkin and dynamical low-rank approximation

Geophysical flow simulations using hyperbolic shallow water moment equations require an efficient discretization of a potentially large system of PDEs, the so-called moment system. This calls for tailored model order reduction techniques that allow for efficient and accurate simulations while guaranteeing physical properties like mass conservation. In this paper, we develop the first model reduction for the hyperbolic shallow water moment equations and achieve mass conservation. This is accomplished using a macro-micro decomposition of the model into a macroscopic (conservative) part and a microscopic (non-conservative) part with subsequent model reduction using either POD-Galerkin or dynamical low-rank approximation only on the microscopic (non-conservative) part. Numerical experiments showcase the performance of the new model reduction methods including high accuracy and fast computation times together with guaranteed conservation and consistency properties.

Moment analysis method for determination of rate constants of solute permeation across interface of spherical molecular aggregates by means of high-performance liquid chromatography

A moment analysis method was developed for the study of solute permeation at the interface of spherical molecular aggregates. At first, new moment equations were developed for determining the partition equilibrium constant (Kp) and permeation rate constants (kin and kout) of solutes from the first absolute (μ1A) and second central (μ2C) moments of elution peaks measured by using high-performance liquid chromatography (HPLC). Then, the method was applied to the analysis of mass transfer phenomena of three solutes, i.e., hydroquinone, resorcinol, and catechol, at the interface of sodium dodecylsulfate (SDS) micelles. HPLC data were measured by using an ODS column and an aqueous phosphate buffer solution (pH = 7.0) as the mobile phase solvent. Pulse response experiments were conducted while changing SDS concentration (5 - 20 mmol dm−3) in the mobile phase under the conditions that the surface of ODS stationary phase was dynamically coated by SDS monomers. In order to demonstrate the effectiveness of the moment analysis method using HPLC, the values of Kp, kin, and kout were determined for the three solutes as 35 - 69, 2.4 × 10−8 - 1.4 × 10−6 m s−1, and 7.0 × 10−10 - 2.1 × 10−8 m s−1, respectively. Their values increase with an increase in the hydrophobicity of the solutes. The method has some advantages for the study of interfacial solute permeation of molecular aggregates. For example, neither immobilization nor chemical modification of both solute molecules and molecular aggregates is required when elution peaks are measured by using HPLC. Interfacial solute permeation takes place in the mobile phase without any chemical reaction or physical action on molecular aggregates. The values of Kp, kin, and kout were analytically determined from those of μ1A and μ2C by using the moment equations. The results of this study must contribute to the dissemination of an opportunity for studying the interfacial solute permeation of molecular aggregates to many researchers because of extremely high versatility of HPLC.

Turbulent flow inside a cubic lid-driven cavity using moment representation lattice Boltzmann method

The present work numerically models the flow inside a cubic lid-driven cavity for Reynolds numbers up to 100 000 using the lattice Boltzmann method. Stable results using the numerical method are obtained, with an implementation of a new set of moment equations for the Dirichlet boundary conditions, allowing approximately one order of magnitude increase in the maximum numerically stable Reynolds number for a given resolution. When evaluating the flow inside the cavity, the flow regime change occurred between Reynolds numbers 20 000 and 25 000, where the core of the turbulent dissipation moves from the bottom of the cavity toward the downstream wall. For Reynolds numbers higher than 50 000, the dissipation was localized near the moving lid. Additionally, negative turbulence production is observed in the bottom wall due to negative velocity gradients caused by the Taylor–Görtler-like vortex colliding with the bottom of the cavity.

Optimization of steady spin prediction by nonlinear iterative inversion algorithm

PurposeSince the inception of aircraft, the phenomenon of spin has persistently accompanied aircraft, and research into spin has been ongoing. This paper aims to introduce an optimization technique to enhance the traditional geometric method for predicting steady spin, aiming to achieve more precise predictive outcomes.Design/methodology/approachTo begin with, the force and moment equations used for motion analysis are initially presented, followed by the establishment of the motion model. Subsequently, the forward problem is set up, and the equilibrium solutions for the left and right spins of the aircraft are determined using the geometric method under the basic and wingtip configurations, thus solving the forward problem. In the final stage, nonlinear inversion is applied, and the inversion objective function is formulated based on the least squares approach. Through iterative processes, the measured data are interpolated, leading to the acquisition of the accurate equilibrium solution.FindingsThe findings indicate that the utilization of the nonlinear iterative inversion algorithm has effectively optimized the geometric method, yielding favorable outcomes. Postoptimization, the prediction accuracy has been enhanced, and the error has significantly diminished when compared to the preoptimization results.Originality/valueThe nonlinear inversion algorithm is used to refine the steady spin prediction for general aviation aircraft. This approach significantly mitigates the precision issues inherent in the forward problem. As demonstrated through the simulations provided, the application of the nonlinear iterative algorithm to resolve the inversion function yields promising optimization outcomes.

Numerical discretisation of hyperbolic systems of moment equations describing sedimentation in suspensions of rod-like particles

We present a numerical discretisation of the coupled moment systems, previously introduced in Dahm and Helzel [1], which approximate the kinetic-fluid model by Helzel and Tzavaras [2] for sedimentation in suspensions of rod-like particles for a two-dimensional flow problem and a shear flow problem. We use a splitting ansatz which, during each time step, separately computes the update of the macroscopic flow equation and of the moment system. The proof of the hyperbolicity of the moment systems in [1] suggests solving the moment systems with standard numerical methods for hyperbolic problems, like LeVeque's Wave Propagation Algorithm [3]. The number of moment equations used in the hyperbolic moment system can be adapted to locally varying flow features. An error analysis is proposed, which compares the approximation with 2N+1 moment equations to an approximation with 2N+3 moment equations. This analysis suggests an error indicator which can be computed from the numerical approximation of the moment system with 2N+1 moment equations. In order to use moment approximations with a different number of moment equations in different parts of the computational domain, we consider an interface coupling of moment systems with different resolution. Finally, we derive a conservative high-resolution Wave Propagation Algorithm for solving moment systems with different numbers of moment equations.

A lightweight, conservation-moment-based implicit gas kinetic Lax–Wendroff scheme for all Knudsen number isothermal gas flows

This work presents an efficient implicit gas kinetic Lax–Wendroff scheme for steady isothermal gas flows in all Knudsen number (Kn) regimes. In the scheme, the discrete velocity Bhatnagar–Gross–Krook model equations (DVE) and the associated conservation moment equations (CME) are coupled and solved by matrix-free implicit schemes. Thanks to obtaining the fluxes of the CME by multiplying the fluxes of the DVE with a projection matrix and utilizing the equilibrium distribution functions at the new time predicted by the CME, both the complicated macro fluxes reconstruction of the CME and the calculation of the Jacobian matrix of the equilibrium distribution functions are not needed in the scheme, which makes the scheme lightweight. Moreover, to enhance the accuracy of the predicted equilibrium distribution functions at the new time to improve the convergence, a symmetric Gauss–Seidel scheme with inner iterations is used to solve the CME system. Due to the coupling between the DVE and the CME, the highly efficient implicit scheme for the CME drives the DVE system to converge quickly for the continuum and near-continuum flows. Furthermore, to verify the accuracy and high efficiency of the proposed implicit scheme, comparison studies of several two-dimensional isothermal rarefied gas flow cases simulated by the present implicit scheme and the explicit gas kinetic Lax–Wendroff scheme are also provided. The numerical results show that the present implicit scheme can be as accurate as its explicit counterpart with one to two orders times speed-up in all Kn number flow regimes.

Variational solution to the lattice Boltzmann method for Couette flow.

Literature studies of the lattice Boltzmann method (LBM) demonstrate hydrodynamics beyond the continuum limit. This includes exact analytical solutions to the LBM, for the bulk velocity and shear stress of Couette flow under diffuse reflection at the walls through the solution of equivalent moment equations. We prove that the bulk velocity and shear stress of Couette flow with Maxwell-type boundary conditions at the walls, as specified by two-dimensional isothermal lattice Boltzmann models, are inherently linear in Mach number. Our finding enables a systematic variational approach to be formulated that exhibits superior computational efficiency than the previously reported moment method. Specifically, the number of partial differential equations(PDEs) in the variational method grows linearly with quadrature order while the number of moment method PDEs grows quadratically. The variational method directly yields a system of linear PDEs that provide exact analytical solutions to the LBM bulk velocity field and shear stress for Couette flow with Maxwell-type boundary conditions. It is anticipated that this variational approach will find utility in calculating analytical solutions for novel lattice Boltzmann quadrature schemes and other flows.

Aerodynamic modeling and performance analysis of model predictive controller for fixed wing vertical takeoff and landing unmanned aerial vehicle

This research focuses mainly on the aerodynamic modelling and performance analysis of a model predictive controller for a hybrid fixed-wing vertical takeoff and landing of unmanned aerial vehicle. The aerodynamics, which comprises several aerodynamic characteristics including the lift, drag, and thrust coefficients, is modelled using Newton’s second law of motion. The force and moment equations were obtained, and they were then converted into matrix and state equation form, together with the transformation matrices. The essential equations with six degrees of freedom (6 DoF) and 12 state matrices including the control input and manipulating variables were obtained. The proposed model predictive controller (MPC) was designed using the optimal model predictive controller design parameters, such as a sampling time of 0.1, a prediction horizon of 15, a control horizon of 3 and additional controller settings. With a settling period of 3 s, an overshoot of 0.5865, and a steady state inaccuracy of 0.00785, the MATLAB simulation demonstrates that the system variables, including roll, pitch, and yaw, are stabilized. This MPC control is more effective in anticipating and optimizing the UAV than other control strategies. Eventually, the controller’s simulation on MATLAB Simulink demonstrates the controller’s ability to stabilize and control the system in a real-time application. Autonomous vertical takeoff and landing operations depend heavily on the mathematical model and architecture of the flight controller. Among the uses are inspection, monitoring, and rescue.