Moment Equations Research Articles

On the use of spherical harmonic approximations in nonclassical particle transport problems

Presented here is the application of the spherical harmonic expansion to the nonclassical transport equation to derive a system of equations for the nonclassical flux moments, namely nonclassical spherical harmonic approximations (NSHA). The nonclassical transport equation is a recently developed mathematical model that allows the modeling of transport problems wherein the particle flux does not undergo exponential attenuation. We employ a Spectral Approach technique to depict the nonclassical flux by expressing it as a truncated series of Laguerre polynomials with respect to the free-path variable s. This yields a system of equations for the nonclassical flux moments structured similarly to the classical PN equations. We show that the NSHA is reduced to the classical PN equation in a special case. Numerical results for slab-geometry test problems are given to verify the derivation.

Capture dynamics and driving method of origami capture mechanism in orbit

This article focuses on the dynamic problem of origami capture mechanism capturing space debris in orbit. It delves into the envelope characteristics of the origami capture mechanism, analyzing the motion laws of its axisymmetric folding state. Furthermore, it establishes a kinematic model for the axisymmetric envelope of the origami capture mechanism. A rigid body dynamic model of the origami capture mechanism and a generalized torque equation for the hinge at the crease were established based on the D–H parameter method and the Lagrange method. The enveloping process of the origami capture mechanism in the microgravity environment was simulated using ADAMS software, and the accuracy of the rigid body dynamics model of the origami capture mechanism and the generalized moment equation of the hinge at the crease was verified. Using Ls-Dyna software to simulate the capture process of space debris by force-driven and displacement-driven flexible origami capture mechanisms in microgravity environments, the results show that displacement-driven origami capture mechanisms can quickly capture space debris and are suitable for capturing large-sized space debris; The force-driven origami capture mechanism can quickly complete envelope capture of space debris, suitable for capturing small-sized space debris.

Topological optimization in 3D-magnetostatics: development of adjoint methods using the equations of magnetic moments

PurposeThe purpose of this paper is mainly to develop the adjoint method within the method of magnetic moment (MMM) and thus, to provide an efficient new way to solve topology optimization problems in magnetostatic to design 3D-magnetic circuits.Design/methodology/approachFirst, the MMM is recalled and the optimization design problem is reformulated as a partial derivative equation-constrained optimization problem where the constraint is the Maxwell equation in magnetostatic. From the Karush–Khun–Tucker optimality conditions, a new problem is derived which depends on a Lagrangian parameter. This problem is called the adjoint problem and the Lagrangian parameter is called the adjoint parameter. Thus, solving the direct and the adjoint problems, the values of the objective function as well as its gradient can be efficiently obtained. To obtain a topology optimization code, a semi isotropic material with penalization (SIMP) relaxed-penalization approach associated with an optimization based on gradient descent steps has been developed and used.FindingsIn this paper, the authors provide theoretical results which make it possible to compute the gradient via the continuous adjoint of the MMMs. A code was developed and it was validated by comparing it with a finite difference method. Thus, a topology optimization code associating this adjoint based gradient computations and SIMP penalization technique was developed and its efficiency was shown by solving a 3D design problem in magnetostatic.Research limitations/implicationsThis research is limited to the design of systems in magnetostatic using the linearity of the materials. The simple examples, the authors provided, are just done to validate our theoretical results and some extensions of our topology optimization code have to be done to solve more interesting design cases.Originality/valueThe problem of design is a 3D magnetic circuit. The 2D optimization problems are well known and several methods of resolution have been introduced, but rare are the problems using the adjoint method in 3D. Moreover, the association with the MMMs has never been treated yet. The authors show in this paper that this association could provide gains in CPU time.

Two-Dimensional System of Moment Equations and Macroscopic Boundary Conditions Depending on the Velocity of Movement and the Surface Temperature of a Body Moving in Fluid

This article is dedicated to the derivation of a two-dimensional system of moment equations depending on the velocity of movement and the surface temperature of a body submerged in fluid, and macroscopic boundary conditions for the system of moment equations approximating the Maxwell microscopic boundary condition for the particle distribution function. The initial-boundary value problem for the Boltzmann equation with the Maxwell microscopic boundary condition is approximated by a corresponding problem for the system of moment equations with macroscopic boundary conditions. The number of moment equations and the number of macroscopic boundary conditions are interconnected and depend on the parity of the approximation of the system of moment equations. The setting of the initial-boundary value problem for a non-stationary, nonlinear two-dimensional system of moment equations in the first approximation with macroscopic boundary conditions is presented, and the solvability of the above-mentioned problem in the space of functions continuous in time and square-integrable in spatial variables is proven.

Developing Closed-Form Equations of Maximum Drag and Moment on Rigid Vegetation Stems in Fully Nonlinear Waves

Developing Closed-Form Equations of Maximum Drag and Moment on Rigid Vegetation Stems in Fully Nonlinear Waves

A reduced-order model for nonlinear radiative transfer problems based on moment equations and POD-Petrov-Galerkin projection of the normalized Boltzmann transport equation

A data-driven projection-based reduced-order model (ROM) for nonlinear thermal radiative transfer (TRT) problems is presented. The TRT ROM is formulated by (i) a hierarchy of low-order quasidiffusion (aka variable Eddington factor) equations for moments of the radiation intensity and (ii) the normalized Boltzmann transport equation (BTE). The multilevel system of moment equations is derived by projection of the BTE onto a sequence of subspaces which represent elements of the phase space of the problem. Exact closure for the moment equations is provided by the Eddington tensor. A Petrov-Galerkin (PG) projection of the normalized BTE is formulated using a proper orthogonal decomposition (POD) basis representing the normalized radiation intensity over the whole phase space and time. The Eddington tensor linearly depends on the solution of the normalized BTE. By linear superposition of the POD basis functions, a low-rank expansion of the Eddington tensor is constructed with coefficients defined by the PG projected normalized BTE. The material energy balance (MEB) equation is coupled with the effective gray low-order equations which exist on the same dimensional scale as the MEB equation. The resulting TRT ROM is structure and asymptotic preserving. A detailed analysis of the ROM is performed on the classical Fleck-Cummings (F-C) TRT multigroup test problem in 2D geometry. Numerical results are presented to demonstrate the ROM's effectiveness in the simulation of radiation wave phenomena. The ROM is shown to produce solutions with sufficiently high accuracy while using low-rank approximation of the normalized BTE solution. Essential physical characteristics of supersonic radiation wave are preserved in the ROM solutions.

Determination of the Speed and Surface Temperature of Aircraft Using the Second Approximation of the System of Moment Equations

This study focuses on solving the initial-boundary value problem for a one-dimensional non-stationary nonlinear system of moment equations in the second approximation. We prove the existence of a unique local in-time solution of the problem under consideration in the space of functions that are continuous in time and square integrable over the spatial variable x. To solve the direct and inverse problems for this system, we develop an iterative numerical method and reduce the system of moment equations to canonical form. Furthermore, we present the software implementation of the algorithms and their application for solving inverse problems related to determining the speed and surface temperature of an aircraft, as well as atmospheric parameters. We provide the results of numerical experiments conducted to validate our approach.

Local intraspecific aggregation in phytoplankton model communities: spatial scales of occurrence and implications for coexistence.

The coexistence of multiple phytoplankton species despite their reliance on similar resources is often explained with mean-field models assuming mixed populations. In reality, observations of phytoplankton indicate spatial aggregation at all scales, including at the scale of a few individuals. Local spatial aggregation can hinder competitive exclusion since individuals then interact mostly with other individuals of their own species, rather than competitors from different species. To evaluate how microscale spatial aggregation might explain phytoplankton diversity maintenance, an individual-based, multispecies representation of cells in a hydrodynamic environment is required. We formulate a three-dimensional and multispecies individual-based model of phytoplankton population dynamics at the Kolmogorov scale. The model is studied through both simulations and the derivation of spatial moment equations, in connection with point process theory. The spatial moment equations show a good match between theory and simulations. We parameterized the model based on phytoplankters' ecological and physical characteristics, for both large and small phytoplankton. Defining a zone of potential interactions as the overlap between nutrient depletion volumes, we show that local species composition-within the range of possible interactions-depends on the size class of phytoplankton. In small phytoplankton, individuals remain in mostly monospecific clusters. Spatial structure therefore favours intra- over inter-specific interactions for small phytoplankton, contributing to coexistence. Large phytoplankton cell neighbourhoods appear more mixed. Although some small-scale self-organizing spatial structure remains and could influence coexistence mechanisms, other factors may need to be explored to explain diversity maintenance in large phytoplankton.

Development of new improved plastic collapse moment equations of pressurized different angled pipe bends under bending moments

In a piping system, pipe bends are more flexible than straight pipes because of their curved geometry, supplemented by higher stress and strain concentration, leading to one of the crucial components in piping industries. Therefore, safe design of pipe bends is essential for smooth running of the piping system, and plastic collapse moment is one of its criteria. This paper utilizes three-dimensional finite element analyses to model empirical solutions for the plastic collapse moment for different angled pipe bends subjected to combined pressure and in-plane closing, in-plane opening, and out-of-plane bending moments. Plastic collapse moments for 30∘ to 180∘ pipe bends are determined for elastic perfectly plastic and strain hardening materials, employing large geometry change option and internal pressure effect. It is observed from results that pressure effect is more prominent in thinner pipe bends of larger bend angle under all bending cases. For in-plane opening and out-of-plane bending moments, collapse moment increases and then decreases with increase in pressure intensity for all sizes of pipe bend. However, for in-plane opening bending moment, collapse moments keep on decreasing for thicker ([Formula: see text] = 11.33) pipe bends. Finally, the study presents new improved plastic collapse moment solutions for different angled pipe bends under bending moment and internal pressure, derived from the finite element results of elastic perfectly plastic and strain hardening material models.

Modal Discontinuous Galerkin Simulations for Grad’s 13 Moment Equations: Application to Riemann Problem in Continuum-Rarefied Flow Regime

For real-world engineering applications, there is a great deal of interest in developing effective models and simulations of continuum-rarefied gas flows. In this study, the numerical simulations of Grad’s 13 (G13) moment equations with application to the Riemann problem in a wide range of continuum-rarefied flow regimes are presented. This work emphasizes numerical robustness and wave phenomena in the G13 system to provide a building block for regularized 13 moment systems, and high-order Grad’s models. For this purpose, a high-order modal discontinuous Galerkin solver is developed for solving one-dimensional G13 moment equations. For spatial discretization, hierarchical modal basis functions premised on orthogonal-scaled Legendre polynomials are used. The proposed approach reduces the G13 systems into a framework of ordinary differential equations in time, which are addressed by an explicit third-order SSP Runge-Kutta algorithm. Three Riemann test cases, including the shock tube, two shock waves and two rarefaction waves, are examined in continuum-rarefied flow regimes. In the G13 system, the arising characteristic waves and dissipation phenomena are investigated in depth. The numerical results demonstrate that every Riemann problem does not have a solution for the G13 system because of loss of hyperbolicity of the system.

The Mechanics of Elastomeric Sheet Reinforced With Bidirectional Fiber Mesh Subjected to Flexure on Boundaries

Abstract We investigate the concurrent three-dimensional deformations of fiber-reinforced composite sheets subjected to out-of-plane bending moments via a continuum model, where we invoke the neo-Hookean strain energy model for the matrix material of fiber-reinforced composite, and assimilate the strain energy of fiber reinforcements into the matrix material model by accounting for stretching, bending, and twisting kinematics of the fibers through the computations of the first-order and second-order gradient of deformation. Emphasis is placed on deriving the Euler equation and boundary conditions of bending moment within the framework of the variational principle and configuring composite surfaces using differential geometry. Significant attention has been given to illustrating the concurrent three-dimensional deformation of fiber composite, meshwork deformation, and fiber kinematics. The simulation results reveal that for a square fiber composite subjected to the out-of-plane bending moment, the maximum in-plane deformation of matrix material occurs along the diagonal direction of the domain while the center of the domain experiences weak in-plane deformation. Notably, the matrix material performs isotropic/anisotropic properties depending on the domain size/shape. In addition, the simulated unit fiber deformations reasonably validate the overall deformation of the network, underscoring that the deformations of the embedded fiber units govern the overall mechanical performance of the fiber meshwork. More importantly, the continuum model qualitatively provides reasonable predictions on the damage patterns of construction materials by demonstrating the kinematics of matrix material and meshwork deformation.

Structural behavior of concrete slabs made of seawater sea sand and reinforced with GFRP reinforcement under flexural loading

Seawater sea sand concrete (SSC) structures reinforced with glass fibre-reinforced polymer (GFRP) reinforcing bars offer a promising alternative to conventional reinforced concrete, especially in the face of depleting clean water and river sand. However, comprehensive studies on the flexural performance and cracking behaviour of SSC structures reinforced with GFRP reinforcing bars are currently limited. This study delves into the influence of longitudinal reinforcement ratio on the flexural and cracking behaviour of SSC slabs reinforced with GFRP reinforcing bars under quasi-static loads, as well as the bond behaviour of GFRP reinforcing bar and SSC. The experimental program included 9 large-scale slabs and 30 pull-out specimens, revealing that the longitudinal reinforcement ratio significantly affects the flexural performance, particularly at higher load levels. Increased reinforcement ratio improved flexural stiffness, reduced deflection and crack width, enhancing load-carrying capacity and energy absorption. Although bond strength decreased at the serviceability load (up to 47%), a marginal reduction was observed at the ultimate state. The study proposes a semi-empirical equation for the effective moment of inertia of SSC slabs, accounting for the influence of reinforcement ratio on cracking moment and reduced bond strength of GFRP reinforcing bars. This equation aligns well with experimental results, demonstrating low error and coefficient of variation.

Radiation hydrodynamics in a moving plasma with Compton scattering: Frequency-dependent solutions

Abstract Radiation hydrodynamical equations with Compton scattering are generally difficult to solve analytically, and usually examined numerically, even if in the subrelativistic regime. We examine the equations available in the subrelativistic regime of kBT$/$(mec2) ≲ 0.1, hν$/$(mec2) ≲ 0.1, and v$/$c ≲ 0.1, where T is the electron temperature, ν the photon frequency, and v the fluid bulk velocity. For simplicity, we ignore the induced scattering terms. We then seek and obtain analytical solutions of frequency-dependent radiative moment equations of a hot plasma with bulk motions for several situations in the subrelativistic regime. For example, in the static case of a plane-parallel atmosphere without bulk motions, where equations involve the generalized Kompaneets equation with subrelativistic corrections, we find the Wien-type solution, which reduces to the usual Milne–Eddington solution in the nonrelativistic limit, as well as the power-law-type one, which has a form of [hν$/$(kBT)]−4. In the moving case of an accelerating one-dimensional flow with bulk motions, we also find the Wien-type and the power-law-type solutions affected by the bulk Compton effect. Particularly, in the Wien-type solutions, due to the bulk Compton effect, the radiation fields gain momentum from the hot plasma in the low-frequency regime of hν < 3kBT, while they lose it in the high-frequency regime of hν > 3kBT.

Identification robust inference for the risk premium in term structure models

We propose identification robust statistics for testing hypotheses on the risk premia in dynamic affine term structure models. We do so using the moment equation specification proposed in Adrian et al. (2013). Statistical inference based on their three-stage estimator requires knowledge of the risk factors’ quality and can be misleading when the β’s are weak, which results when sampling errors are of comparable order of magnitude as the risk factor loadings. We extend the subset (factor) Anderson–Rubin test from Guggenberger et al. (2012) to models with multiple dynamic factors and time-varying risk prices. It provides a computationally tractable manner to conduct identification robust tests on a few risk premia when a larger number is present. We use it to analyze potential identification issues arising in the data from Adrian et al. (2013) for which we show that some factors, though potentially weak, may drive the time variation of risk prices, and weak identification issues are more prominent in multi-factor models.

Долгосрочная оценка режима минимального зимнего стока рек бассейна озера Севан при изменении климата

Introduction. Analysis and assessment of the statistically significant consequences of climate change for the long-term minimum 30-day winter runoff of rivers in the Lake Sevan basin is of significant interest to consumers and water resource management, being an important hydrological issue. The relevance of consideration of the issue is due to the fact that for Armenia Lake Sevan is a strategic reservoir of fresh water, a large water management complex, a source of water for hydropower and agricultural irrigation, an object of tourism and recreation. Issues of formation and calculation of the minimum runoff of rivers in the lake basin. Sevan were discussed in works, the analysis of the spatial-temporal distribution of the winter minimum decade-day, monthly and 30-day runoff can be found, for example, in works.In contrast to the listed works, in this study a technique for assessing the statistically significant consequences of climate change for the hydrological regime in relation to small foothill watersheds of the rivers of the Lake Sevan basin (Republic of Armenia) is considered. The values of the minimum 30-day water runoff within the study area can be used when calculating standards for permissible discharges into rivers, to assess natural groundwater resources, as indicators of the extremeness of river flow and in other water management calculations. Research methods and materials. The methodology is based on a probabilistic model of the formation of long-term river runoff, which is approximated by a system of linear equations for the initial statistical moments. As the source material were used the data of actual hydrometeorological observations (minimum 30-day winter river runoff, air temperature and precipitation) of the “Center for Hydrometeorology and Monitoring” of the SNCO of the Ministry of Environment of the Republic of Armenia. Research results. The results of approbation of the methodology on retrospective material are presented, taking into account possible climate change both towards warming and towards cooling.The conditions for the applicability of the methodology to small foothill watersheds for assessing the minimum 30-water winter river runoff are identified. The obtained satisfactory results made it possible to assess the impact of climate change for the minimum runoff using the IPCC climate scenarios. Three climatic scenarios are considered, the model of the Max Planck Institute with a monthly time resolution. Conclusions. As a result, watersheds of rivers in the basin of Lake Sevan were identified, in which significant deviations of the predicted average annual values of the minimum 30-day winter runoff and the coefficients of its variation from the actual ones in the implementation of the considered scenarios can be expected. An analysis of the obtained results is given from the point of view of taking possible preventive measures, including strategic planning of the use of water resources in winter low water, in case of climate change.

Developing Dynamic System Responses Model Based on Nanotechnology by Incorporating the Finite Element Analysis

The dynamic model of complex structure has set incredible progression with the use of finite element analysis (FEA), especially in the field of vibration control and nanotechnology. Main focus of this research is to conduct a frequency analysis in ANSYS and a state space model of the beam under the platform of MATLAB. This allowed a system to develop an active dynamic model for vibration control. To enhance test and analysis of the cantilever beam of Aluminium 6061T6, a new method has been proposed to predict the dynamic response of a cantilever beam under sinusoidal base excitation. This was achieved by creating an analytical model for the cantilever beam using moment and force equilibrium equations. The authenticity of the proposed method was made by comparing the results with experimental data. Additionally, to control vibration, a Proportional-Integral-Derivative (PID) controller was developed using the system model. The FEA in the ANSYS platform provided a cantilever beam model. Also, its mathematical modelling has been done. The proposed method utilizes a novel disturbance rejection control scheme that eliminates an unknown disturbance. Experimental results indicate that the control system results in gradually decreased beam vibration amplitude. The state space approach discussed in the work could be a valuable tool for studying the behaviours of nanomaterials.

Equilibrium analysis of surface-constrained elastic rods: Unveiling contact and internal forces through local geometry

Confinement scenarios of thin elastic rods are prevalent in both natural and engineered systems. The accurate quantification of the mechanical interplay between confined rods and their confining surfaces remains a formidable challenge, primarily due to the intricate nonlinear nature of thin rods and their contact with surfaces. Here, we present a theoretical framework designed to characterize equilibrium states of thin elastic rods constrained to general surfaces, accounting for the influence of both conservative force and rod inhomogeneity. The framework proposes a general scheme to directly unveil contact and internal forces of confined rods through the consideration of local geometry, based on balance equations for local forces and moments. Characteristics of the contact force are contingent upon whether the deformed rod exhibits vanishing geodesic curvature. Case studies involving closed rods constrained to spherical, cylindrical, and torus surfaces are conducted to elucidate geometric effects on various rod behavior, encompassing equilibrium configurations, internal forces and moments, and contact forces. The outcomes of our study offer fundamental insights into how geometric and material properties lead to the intricate mechanical interactions between one- and two-dimensional materials.

Polarization dynamics from moment equations

Polarization dynamics from moment equations

Stochastic kinetic theory applied to coarse-grained polymer model.

A stochastic field theory approach is applied to a coarse-grained polymer model that will enable studies of polymer behavior under non-equilibrium conditions. This article is focused on the validation of the new model in comparison with explicit Langevin equation simulations under conditions with analytical solutions. The polymers are modeled as Hookean dumbbells in one dimension, without including hydrodynamic interactions and polymer-polymer interactions. Stochastic moment equations are derived from full field theory. The accuracy of the field theory and moment equations is quantified using autocorrelation functions. The full field theory is only accurate for a large number of polymers due to keeping track of rare occurrences of polymers with a large stretch. The moment equations do not have this error because they do not explicitly track these configurations. The accuracy of both methods depends on the spatial degree of discretization. The timescale of decorrelation over length scales bigger than the spatial discretization is accurate, while there is an error over the scale of single mesh points.

Moment Analysis Method for Measurement of Reaction Equilibrium and Rate Constants by Using High-Performance Liquid Chromatography.

The moment analysis method was developed for the determination of association equilibrium constant (KA) and association (ka) and dissociation (kd) rate constants of intermolecular interactions between solute and ligand molecules. They are accurately determined by using moment equations from elution peak profiles because they are measured by using high-performance liquid chromatography (HPLC) under preferable conditions that neither immobilization nor chemical modification (i.e., fluorescence labeling) of solute and ligand molecules is required. To demonstrate the effectiveness of the method, it was applied to the inclusion complex formation system between dibenzo-18-crown-6 (DB18C6) and alkaline earth metal cations, i.e., Mg2+, Ca2+, and Sr2+, as a concrete example. Because the diameter of the three metal cations is smaller than that of the inner cavity of DB18C6, the values of KA, ka, and kd were analytically determined by assuming the stoichiometry of 1:1 between DB18C6 and the metal cation. They reflected the influence of the difference in the size between the inner cavity of DB18C6 and the metal cations on the inclusion complex formation. It seems that the moment analysis method based on HPLC separation is effective for the multifaceted analysis of chemical reactions because some characteristics of the method are different from those of other conventional methods. It must contribute to the dissemination of an opportunity for the study of chemical reactions to many researchers because of the versatility of HPLC.