Energy Conservation Law Research Articles

Lie group symmetry analysis and invariant difference schemes of the two-dimensional shallow water equations in Lagrangian coordinates

The two-dimensional shallow water equations in Lagrangian coordinates are considered. Lie group classification for the class of the elliptic paraboloid bottom topography is performed.The transformations mapping the two-dimensional shallow water equations with a plane or rotation symmetric bottom into the gas dynamics equations of a polytropic gas with polytropic exponent γ=2 are represented. The group foliation of the two-dimensional shallow water equations in Lagrangian coordinates is discussed.New invariant conservative finite-difference schemes for the equations and their one-dimensional reductions are constructed. The schemes are derived either by extending the known one-dimensional schemes or by direct algebraic construction based on some assumptions on the form of the energy conservation law. Among the proposed schemes there are schemes possessing conservation laws of mass and energy.

A new rolling friction coefficient measurement method based on the work-energy theorem

The rolling friction coefficient represents a characteristic parameter for the rolling motion. To determine the rolling friction coefficient, we propose a simple method based on the conservation of energy. The measuring setup includes the tools and materials that are simple and easy to obtain, such as the spring, ruler, and a small laboratory wheeled cart. In this paper, we have determined the rolling friction coefficient for several masses of the test carts and different lengths of spring compression. When the spring is compressed, its energy is directly proportional to the square of the length of the spring compression. The initial speed of the cart was determined using the law of energy conservation, and the length that the cart goes before stopping is measured with a ruler. The conclusion is that the value of the rolling friction coefficient is usually very small and that the mass of an object will affect the friction force but will not affect the rolling friction coefficient. Besides determining the rolling friction coefficient, this technique should help students comprehend the concept of friction easily.

Optimal point-wise error estimates of two conservative finite difference schemes for the coupled Gross–Pitaevskii equations with angular momentum rotation terms

This paper is concerned with numerical analysis of two finite difference schemes for solving the coupled Gross–Pitaevskii equations with angular momentum rotation terms which model the dynamics of rotating two-component Bose–Einstein condensates. Both schemes are proved to be uniquely solvable and inherit the conservation laws of mass and energy in the discrete sense. Besides the energy method, the ‘cut-off’ function technique and inductive argument are introduced to establish the optimal error estimates of the numerical methods without any requirement on the grid-ratio or initial values. The convergence order of the numerical solutions to the exact solutions is proved to be of O(τ2+h2) with time-step τ and mesh size h. Several numerical results are reported to validate the error estimates and conservation laws.

Universal bounds on quantum mechanics through energy conservation and the bootstrap method

Abstract The range of motion of a particle with certain energy E confined in a potential is determined from the energy conservation law in classical mechanics. The counterpart of this question in quantum mechanics can be regarded as what is the possible range of expectation values of the position operator 〈x〉 of a particle that satisfies E = 〈H〉. This range depends on the state of the particle, but the universal upper and lower bounds, which are independent of the state, must exist. In this study, we show that these bounds can be derived by using the bootstrap method. We also point out that the bootstrap method can be regarded as a generalization of the uncertainty relations, meaning that the bounds are determined by the uncertainty relations in a broad sense. Furthermore, the bounds on possible expectation values of various quantities other than position can be determined in the same way. However, in the case of multiple identical particles (bosons and fermions), we find some difficulty in the bootstrap method. Because of this issue, the predictive power of the bootstrap method in multi-particle systems is limited in the derivation of observables including energy eigenstates. In addition, we argue an application of the bootstrap method to thermal equilibrium states. We find serious issues that temperature and entropy cannot be handled. Although we have these issues, we can derive some quantities in micro-canonical ensembles of integrable systems governed by generalized Gibbs ensembles.

Structure-preserving scheme for one dimension and two dimension fractional KGS equations

<abstract><p>In the paper, we study structure-preserving scheme to solve general fractional Klein-Gordon-Schrödinger equations, including one dimension case and two dimension case. First, the high central difference scheme and Crank-Nicolson scheme are used to one dimension fractional Klein-Gordon-Schrödinger equations. We show that the arising scheme is uniquely solvable, and approximate solutions converge to the exact solution at the rate $ O(\tau^2+h^4) $. Moreover, we prove that the resulting scheme can preserve the mass and energy conservation laws. Second, we show Crank-Nicolson scheme for two dimension fractional Klein-Gordon-Schrödinger equations, and the proposed scheme preserves the mass and energy conservation laws in discrete formulations. However, the obtained discrete system is nonlinear system. Then, we show a equivalent form of fractional Klein-Gordon-Schrödinger equations by introducing some new auxiliary variables. The new system is discretized by the high central difference scheme and scalar auxiliary variable scheme, and a linear discrete system is obtained, which can preserve the energy conservation law. Finally, the numerical experiments including one dimension and two dimension fractional Klein-Gordon-Schrödinger systems are given to verify the correctness of theoretical results.</p></abstract>

Structures, thermodynamics and dynamics of topological defects in Gay-Berne nematic liquid crystals.

Topological defects are a ubiquitous phenomenon across different physical systems. A better understanding of defects can be helpful in elucidating the physical behaviors of many real materials systems. In nematic liquid crystals, defects exhibit unique optical signatures and can segregate impurities, showing their promise as molecular carriers and nano-reactors. Continuum theory and simulations have been successfully applied to link static and dynamical behaviors of topological defects to the material constants of the underlying nematic. However, further evidence and molecular details are still lacking. Here we perform molecular dynamics simulations of Gay-Berne particles, a model nematic, to examine the molecular structures and dynamics of +1/2 defects in a thin-film nematic. Specifically, we measure the bend-to-splay ratio K3/K1 using two independent, indirect measurements, showing good agreement. Next, we study the annihilation event of a pair of ±1/2 defects, of which the trajectories are consistent with experiments and hydrodynamic simulations. We further examine the thermodynamics of defect annihilation in an NVE ensemble, leading us to correctly estimate the elastic modulus by using the energy conservation law. Finally, we explore effects of defect annihilation in regions of nonuniform temperature within these coarse-grained molecular models which cannot be analysed by existing continuum level simulations. We find that +1/2 defects tend to move toward hotter areas and their change of speed in a temperature gradient can be quantitatively understood through a term derived from the temperature dependence of the elastic modulus. As such, our work has provided molecular insights into structures and dynamics of topological defects, presented unique and accessible methods to measure elastic constants by inspecting defects, and proposed an alternative control parameter of defects using temperature gradient.

Confirmation of the First Law of Thermodynamics in Theory and Extended Bernoulli Equation

The internal energy change of ideal gas does not depend on the volume and pressure. The internal energy change of real gas has not any relation with the volume and pressure, which had been proved. If the internal energy change had not any relation with the volume and pressure, we could confirm the first law of thermodynamics in theory. Simultaneously, the internal energy change is the state function that shall be able to be proved in theory. If the internal energy change depended on the volume and pressure, we could not prove that the internal energy change is the state function and the chemical thermodynamics theory is right. The extended or modified Bernoulli equation can be derived from the energy conservation law, and the internal energy change, heat, and friction are all considered in the derivation procedure. The extended Bernoulli equation could be applied to the flying aircraft and mechanical motion on the gravitational field, for instance, the rocket and airplane and so on. This paper also revises some wrong ideas, viewpoints, or concepts about the thermodynamics theory and Bernoulli equation.

Theoretical preconditions for the development of mathematical models of the technology of desert plant drying

The formulation of stationary and non-stationary problems for drying dispersed materials is substantiated in the article. A mathematical model and its analytical solutions are developed to determine the patterns of the interdependence of temperature and moisture content. A mathematical model of the process of drying a dispersed material is based on the energy conservation law under heat and mass transfer in a multiphase medium.

Investigation of monitoring tasks of the dynamics of heating networks temperature states

In order to monitor heat supply systems for the purpose of uninterrupted heat supply to consumers, it is necessary to create full-scale monitoring systems that would improve the quality of decisions on controlling the development and functioning of systems based on the information coming from continuously operating measuring systems. The main result and purpose of monitoring in this case will be the assessment of the current regime and its compliance with the established requirements. The article investigates the task of monitoring the dynamics of temperature states of heating networks. As the closing equation in the thermal-hydraulic model, it is proposed to use the equation based on the energy conservation law, describing physical processes related to heat transfer through the walls of pipelines and heat transfer into the environment.

Design of falling rock preventive countermeasure based on three-dimensional simulation

To rationally design rockfall countermeasures, detailed rockfall behavior must be predicted using numerical simulations based on the conditions of slopes and rockfalls. In this study, we perform a rockfall simulation using discontinuous deformation analysis. This method is used to analyze the motion between blocks composed of polygons of arbitrary elastic bodies; furthermore, the energy conservation law and the uniqueness/convergence of the solution are guaranteed for the entire analysis target. Additionally, the rockfall shape is reproduced in detail. Both the rockfall shape and a parameter known as the velocity energy ratio, which indicates the decay of energy at the collision time of the rockfall, significantly affect the rockfall behavior. In this study, we model the actual slope shape from data measured via airborne laser and quantitatively evaluate the effect via simulations by changing the rockfall shape and velocity energy ratio. In this paper, we consider how to use the simulation for the design of rockfall countermeasures based on the results such as the change in the amount of jumping due to the difference in slope shape, geology, and rockfall shape.

Конструирование релятивистской гидродинамики многокомпонентной жидкости. 1. Метод релятивистской необратимой термодинамики

The paper develops relativistic mechanics and irreversible thermodynamics of a cosmological liquid mixture in which thermal conduction, diffusion, viscous flow and their crossing phenomena can occur. The main thermodynamic fields that occur in relativistic irreversible thermodynamics are defined as statistical expressions using relativistic kinetics. The effects of the Eckart approach on the choice of the hydrodynamic velocity are shown. The equations of relativistic multicomponent hydrodynamics for local densities of momentum, energy and number of particles of different sort, are also obtained from the fundamental conservation laws of relativistic thermodynamics.. The energy conservation law (the first beginning of relativistic thermodynamics) is formulated. Covariant Gibbs relation and local form of the second principle of thermodynamics in the presence of entropy source were derived in order to obtain determining relations linearly connecting fluxes with corresponding thermodynamic forces. Equivalent forms of relativistic entropy production in the form of a bilinear expression in terms of thermodynamic forces were obtained. A new cross-effect between diffusion and conduction arising from the relativistic term in the thermodynamic force coupled to the heat flux is discussed. The synopsis presented is of interest for relativistic mechanics, astrophysics, and cosmology.

Simplified Calculation of Lateral Displacement of a Cement‐Soil Gravity Retaining Wall

A common retaining structure in Shanghai’s soft soil foundation pit is the cement‐soil gravity retaining wall. In this research, a streamlined calculation method is suggested to examine the lateral displacement of the cement‐soil gravity retaining wall’s deformation features. First, the lateral pressure of the retaining wall in a nonlimit passive state is calculated by separately estimating water and soil pressure. Then, the work performed by the lateral pressure of the retaining wall and the internal energy of deformation is calculated and analyzed according to the law of energy conservation to obtain the maximum lateral displacement at the top of the retaining wall. At last, the influences of wall insertion ratio, wall width, cement mixing ratio, and soil internal friction angle on the lateral displacement of retaining wall are analyzed with finite element numerical simulation. The results of the streamlined calculations are contrasted with the measured findings when paired with engineering scenarios, which demonstrates that the predicted lateral displacement is reasonably accurate with an average error of less than 15%.

EVALUATION OF DISSIPATION ENERGY OF ISOTROPIC CONTINUUM DAMAGE MECHANICS MODEL WITH ADAPTIVE TIME-STEP CONTROL APPROACH

This paper proposes an algorithm to evaluate dissipation energy of an isotropic continuum damage mechanics model with an adaptive time-step control approach. The algorithm takes place at each integration point under the scope of finite element analysis. The total amount of the dissipation energy of one structure can be used to verify the conservation law of energy, where the summation of the elastic strain energy and the dissipation energy should be equal to the total work by external force. However, the paper shows that this condition may not be valid when the unstable crack propagation occurs. When the crack propagation is unstable, the development of damage can be driven by stored strain energy without external load. The strains at the beginning and ending time within one increment at the damaged elements will give inaccurate dissipation energy by the classic trapezoidal integration scheme. In addition, this paper proposes a new damage evolution shape function with C1 continuity of the strain-stress curve under the uniaxial tension test, which is used to demonstrate the numerical procedure of the adaptive algorithm.

Напівемпіричне оцінювання акустичних навантажень на головну частину ракети при нестандартній конфігурації стартових споруд

The paper investigates the possibility of using combinations of two options of wave fields conjugation (the conjugation by points and the root mean square approximation) in the method of partial domains using the example of solving the problem of piston radiation in a finite wedge-shaped waveguide. It is shown that the option of using the root mean square approximation is the best, while the use of the conjugation by points of fields generally simplifies the calculation of the problem. The choice of the required calculation combination depends on the problem under consideration, and it is possible that the combined calculation option may lead to a better solution to the problem in terms of fulfilling the law of energy conservation or field conjugation conditions.

A linearly implicit energy-preserving exponential time differencing scheme for the fractional nonlinear Schrödinger equation

<abstract><p>In this paper, we present a new method to solve the fractional nonlinear Schrödinger equation. Our approach combines the invariant energy quadratization method with the exponential time differencing method, resulting in a linearly-implicit energy-preserving scheme. To achieve this, we introduce an auxiliary variable to derive an equivalent system with a modified energy conservation law. The proposed scheme uses stabilized exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space to obtain a linearly-implicit, fully-discrete scheme. Compared to the original energy-preserving exponential integrator scheme, our approach is more efficient as it does not require nonlinear iterations. Numerical experiments confirm the effectiveness of our scheme in conserving energy and its efficiency in long-time computations.</p></abstract>

Prediction of Peak Overpressure of Charge Enveloped by Polymer Matrix Composite: Theoretical Modeling and Experimental Verification.

This study aimed at elucidating some characteristics of the shock wave overpressure generated by a non-traditional layered charge comprising an inner high-energy explosive and an outer polymer matrix composite. Two models for predicting the peak overpressure (Δpm) of the charge were established, namely, a model based on the initial parameters of the blast wave, and a model considering the weakening of the explosion energy through the introduction of polymer matrix cladding. The overpressure of a typical layered charge was experimentally measured for model validation. It was found that the difference between the Δpm predicted by the two models and the experimental data is less than 15.12% and 14.17%, respectively. The model that was established based on the conservation of energy law, is in best agreement with the experimental data under different cladding/charge mass ratios (αm). The model that was based on the initial parameters of the blast wave obtained a low predicted value when αm was 0.4-0.8, which is attributed to the non-uniformity of the gas-solid mixture during the explosive dispersion stage.

Numerical simulation of selective laser melting by the SPH method

Abstract. Currently, additive manufacturing technologies develop actively. This requires creation of computational methods to describe physical processes occurring at the time of manufacturing. One of the methods used for the production of metal powder parts is the method of selective laser melting. This paper presents an SPH-based numerical technique for modeling the process of powder sintering under the influence of a laser beam. The flow of liquid formed as a result of melting is described by the Navier-Stokes equations. Pressure forces, viscous effects and surface forces at the interface are included in the force balance. The thermal state is determined from the energy conservation law, which takes into account thermal processes, volumetric absorption of laser radiation energy, convective heat exchange with the external environment and radiation. Phase transitions between solid and liquid phases are described in the framework of the generalized formulation of the Stefan problem. The calculation method is verified on tests specific to the class of problems under consideration. A comparison is made with the analytical solution, as well as with solutions obtained by other modifications of the SPH method, and with experimental data.

Non-standard analysis in electrical engineering. Transient analysis in second-order electrical circuits with violation of switching laws

For the first time, the authors proposed the use of the mathematical apparatus of non-standard analysis to solve certain non-trivial problems of theoretical electrical engineering. It was established that the axiomatics of non-standard analysis allows to simplify the analysis of transient processes in high-order electric circuits when the switching laws are violated. It is shown that the application of non-standard analysis methods in theoretical electrical engineering provides an opportunity to use the traditional classical method of transient processes analysis of circuits with violation of switching laws. Only by using methods of non-standard analysis, it is possible to strictly prove the fulfillment of the law of energy conservation in such schemes. Also, examples of solving such tasks are given. It is recommended to expand the scope of application of non-standard methods of analysis in problems of various branches of science and technology, which use differential calculus and boundary transitions, and the solution of which is limited or impossible by standard approaches.

Quantitative characterization of volcanic expansion fractures based on thermodynamic coupling analysis

In recent years, volcanic reservoirs in the Bohai Sea have become the focus of oil and gas exploration and development. Volcanic activity during the Cenozoic provided favorable conditions for the development of hydrocarbon traps. These volcanics also result in difficulties during the drilling and development of oil and gas fields. Furthermore, the high-temperature magma induced thermal expansion in the surrounding rocks, which changed the local stress field and affected the fracture distribution characteristics of the surrounding strata. The complexity of the fracture distribution makes predicting lost circulation difficult and causes substantial economic losses. Therefore, it is necessary to characterize the volcanic perimeter expansion-type fractures effectively. The distribution of volcanic rocks is identified through logging and seismic data, with perimeter fracture development characteristics being analyzed. The temperature conditions of the volcanic activity were determined through core mineral analysis and thin section observation. Based on the laws of energy conservation and geomechanics, the principle of thermal coupling was applied to quantitatively predict the thermal expansion fractures caused by volcanic activity using a finite element numerical simulation method. To determine the extent of volcanic influence on lost circulation, and the development characteristics of fractures under the influence of these rocks were analyzed. The results provide an important reference for drilling and volcanic reservoir prediction in the Bohai Basin and guide for elsewhere.

Two-way wave–vortex interactions in a Lagrangian-mean shallow water model

We derive and investigate numerically a reduced model for wave–vortex interactions involving non-dispersive waves, which we study in a two-dimensional shallow water system with an eye towards applications in atmosphere–ocean fluid dynamics. The model consists of a coupled set of nonlinear partial differential equations for the Lagrangian-mean velocity and the wave-related pseudomomentum vector field defined in generalized Lagrangian-mean theory. It allows for two-way interactions between the waves and the balanced flow that is controlled by the distribution of Lagrangian-mean potential vorticity, and for strong solutions it features a desirable exact energy conservation law for the sum of wave energy and mean flow energy. Our model goes beyond standard ray tracing as we can derive weak solutions that contain discontinuities in the pseudomomentum field, using the theory of weakly hyperbolic systems. This allows caustics to form without predicting infinite wave amplitudes, as would be the case in the standard ray-tracing theory. Suitable wave forcing and dissipation terms are added to the model and a numerical scheme for the model is implemented as a coupled set of pseudo-spectral and finite-volume integrators. Idealized examples of interactions between wavepackets and simple vortex structures are presented to illustrate the model dynamics. The unforced and non-dissipative simulations suggest a heuristic rule of ‘greedy’ waves, i.e. in the long run the wave field always extracts energy from the mean flow.