Conservation Laws Research Articles

Unifying finite-temperature dynamical and excited-state quantum phase transitions

In recent years, various notions of dynamical phase transitions have emerged to describe far-from-equilibrium criticality. A unifying framework connecting these different concepts is still missing, and would provide significant progress toward understanding far-from-equilibrium quantum many-body universality. Initializing our system in a thermal ensemble and subsequently performing quantum quenches in the Lipkin-Meshkov-Glick model, we establish a direct connection between excited-state quantum phase transitions (ESQPTs) and two major types of dynamical phase transitions (DPTs), by relating the phases of the latter to the critical energies and conservation laws in the former. Our work provides further insight into how various concepts of non-ground-state criticality are intimately connected, paving the way for a unified framework of far-from-equilibrium universality. Published by the American Physical Society 2024

Full-scale experimental study on the smoke descent and stratification in a two-storey building with varying ventilation conditions

This paper presents a full-scale experimental research on the smoke descent and stratification in a two-storey residential building under the coupling influences of vertical connection channels and ventilation conditions. The two floors were connected by one atrium and one stair, and the fire source was located on the second floor. Wood crib and gasoline were selected as representative fuels to reflect typical residential fire scenarios, with heat release rates of 82kW and 140kW, respectively. The states of the building openings were varied to simulate different ventilation conditions. Results showed that the smoke tended to spread in the ridge direction due to the structural limitations of slope roofs. The temperature distribution in both the ridge and slope directions was obtained. A special “three-zone” smoke stratification structure including upper smoke layer, intermediate air layer and lower smoke layer is observed, which is significantly different from the well-known “two-zone” model. The “three-zone” stratification resulted in a nonmonotonic vertical temperature distribution in the atrium. The lower smoke layer thickness Zl,g was found to be greatly affected by ventilation conditions. Based on the boundary layer theories and the law of energy conservation, a series of expressions were established to calculate Zl,g and intermediate air layer thickness δw, which are the key parameters of the “three-zone” stratification. The theoretical calculations matched the experimental data well and explained the sharp trends of δw and Zl,g under different ventilation conditions. This study can provide theoretical guidance about fire detection, smoke exhaust and people evacuation in residential buildings.

Enhanced Attractive Force via Torsion Spring as Rotatable Tip Structure in Beam-Structure Electrostatic Chucks

This research aims to replace ball joints of beam-structure electrostatic chucks (BSESC), which are limited by issues such as structural friction and inconsistent operation, thus enhancing the applicability in industrial settings. Torsion springs are used as rotational degree-of-freedom (RDOF) mechanisms at the tip of BSESCs to reduce friction and improve repeatability, while also highlighting the positive relationship between tip rotatability and the attractive force generated. Following the law of conservation of energy, an analytical calculation was performed using a beam-spring-capacitor model to establish the correlation between tip rotatability and the maximum attractive force generated through energy system analysis. An analytical solution was derived, explicitly defining the relationship between tip rotatability and the achievable attractive force. The results were plotted using MATLAB. Experimental validations were carried out by fabricating devices that differed only in tip rotatability, and measurements of the attractive force were taken. Both analytical and experimental results clearly demonstrate a positive relationship between tip rotatability and attractive force generation in BSESCs. The use of torsion springs effectively reduces the energy concentration at the tip, delaying detachment while simultaneously increasing the attractive force, significantly improving the overall performance of BSESCs. These findings provide a foundation for future industrial applications, particularly for automated large-area manipulation of irregularly shaped targets in industries such as screen production and semiconductors.

A model of freeze desalination for predicting salt concentration in ice using the laws of conservation of matter & mass

The assessment of freeze desalination (FD) efficiency is lagging behind due to the lack of an effective means of determining ice total dissolved solid (TDS). In this paper, an ice TDS prediction model was constructed by utilizing the law of conservation of matter (LCMat) and conservation of mass (LCMas) in the icing process. It was concluded that the average TDS ice crystals in any icing process was correlated with the TDS of the raw (concentrated) water at the starting of freezing, the salt mass change rate of the concentrate water, and the volume of ice crystals. A one-way FD test was carried out using the saltwater from South Xinjiang, and the exponential equation model and linear equation model of LCMat-LCMas were derived, and the simulation accuracy of LCMat-LCMas was verified by using the test data. The results show that the model has high prediction accuracy. The exponential equation model is suitable for predicting the ice TDS for ice with thickness within 15 cm, and the linear equation is suitable for predicting the ice crystal TDS for ice thicker than 15 cm. Further validation of the model accuracy is needed for larger cooled areas and icing thicknesses, and multidirectional icing processes.

Discretely Nonlinearly Stable Weight-Adjusted Flux Reconstruction High-Order Method for Compressible Flows on Curvilinear Grids

To achieve genuine predictive capability, an algorithm must consistently deliver accurate results over prolonged temporal integration periods, avoiding the unwarranted growth of aliasing errors that compromise the discrete solution. Provable nonlinear stability bounds the discrete approximation and ensures that the discretization does not diverge. Nonlinear stability is accomplished by satisfying a secondary conservation law, namely for compressible flows; the second law of thermodynamics. For high-order methods, discrete nonlinear stability and entropy stability, have been successfully implemented for discontinuous Galerkin (DG) and residual distribution schemes, where the stability proofs depend on properties of L2-norms. In this paper, nonlinearly stable flux reconstruction (NSFR) schemes are developed for three-dimensional compressible flow in curvilinear coordinates. NSFR is derived by merging the energy stable flux reconstruction (ESFR) framework with entropy stable DG schemes. NSFR is demonstrated to use larger time-steps than DG due to the ESFR correction functions, at the cost of larger error levels at design order convergence for equivalent degrees of freedom, while preserving discrete nonlinear stability. NSFR differs from ESFR schemes in the literature since it incorporates the FR correction functions on the volume terms through the use of a modified mass matrix. We also prove that discrete kinetic energy stability cannot be preserved to machine precision for quadrature rules where the surface quadrature is not a subset of the volume quadrature. This result stems from the inverse mapping from the kinetic energy variables to the conservative variables not existing for the kinetic energy projected variables. This paper also presents the NSFR modified mass matrix in a weight-adjusted form. This form reduces the computational cost in curvilinear coordinates because the dense matrix inversion is approximated by a pre-computed projection operator and the inverse of a diagonal matrix on-the-fly and exploits the tensor product basis functions to utilize sum-factorization. The nonlinear stability properties of the scheme are verified on a nonsymmetric curvilinear grid for the inviscid Taylor-Green vortex problem and the correct orders of convergence were obtained on a curvilinear mesh for a manufactured solution. Lastly, we perform a computational cost comparison between conservative DG, overintegrated DG, and our proposed entropy conserving NSFR scheme, and find that our proposed entropy conserving NSFR scheme is computationally competitive with the conservative DG scheme.

The influence of parasitic modes on stable lattice Boltzmann schemes and weakly unstable multi-step Finite Difference schemes

Numerical analysis for linear constant-coefficient multi-step Finite Difference schemes is a longstanding topic, developed approximately fifty years ago. It relies on the stability of the scheme, and thus—within the L2 setting—on the absence of multiple roots of the amplification polynomial on the unit circle. This allows for the decoupling, while discussing the convergence of the method, of the study of the consistency of the scheme from the precise knowledge of its parasitic/spurious modes, so that the methods can be essentially studied as if they had only one step. Furthermore, stability alleviates the need to delve into the complexities of floating-point arithmetic on computers, which can be challenging topics to address. In this paper, we demonstrate that in the case of “weakly” unstable Finite Difference schemes with multiple roots on the unit circle, although the schemes may remain stable, considering parasitic modes is essential in studying their consistency and, consequently, their convergence. This research was prompted by unexpected numerical results on stable lattice Boltzmann schemes, which can be rewritten in terms of multi-step Finite Difference schemes. Unlike Finite Difference schemes, rigorous numerical analysis for lattice Boltzmann schemes is a contemporary topic with much left for future discoveries. Initial expectations suggested that third-order initialization schemes would suffice to maintain the accuracy of fourth-order schemes. However, this assumption proved incorrect for weakly unstable Finite Difference schemes and for stable lattice Boltzmann methods. This borderline scenario underscores that particular care must be adopted for lattice Boltzmann schemes, and the significance of genuine stability in facilitating the construction of Lax-Richtmyer-like theorems and in mastering the impact of round-off errors concerning Finite Difference schemes. Despite the simplicity and apparent lack of practical usage of the linear transport equation at constant velocity considered throughout the paper, we demonstrate that high-order lattice Boltzmann schemes for this equation can be used to tackle nonlinear systems of conservation laws relying on a Jin-Xin approximation and high-order splitting formulæ.

Physics-informed neural networks for weakly compressible flows using Galerkin–Boltzmann formulation

In this work, we study the Galerkin–Boltzmann formulation within a physics-informed neural network (PINN) framework to solve flow problems in weakly compressible regimes. The Galerkin–Boltzmann equations are discretized with second-order Hermite polynomials in microscopic velocity space, which leads to a first-order conservation law with six equations. Reducing the output dimension makes this equation system particularly well suited for PINNs compared with the widely used D2Q9 lattice Boltzmann velocity space discretizations. We created two distinct neural networks to overcome the scale disparity between the equilibrium and non-equilibrium states in collision terms of the equations. We test the accuracy and performance of the formulation with benchmark problems and solutions for forward and inverse problems with limited data. We compared our approach with the incompressible Navier–Stokes equation and the D2Q9 formulation. We show that the Galerkin–Boltzmann formulation results in similar L2 errors in velocity predictions in a comparable training time with the Navier–Stokes equation and lower training time than the D2Q9 formulation. We also solve forward and inverse problems for a flow over a square, try to capture an accurate boundary layer, and infer the relaxation time parameter using available data from a high-fidelity solver. Our findings show the potential of utilizing the Galerkin–Boltzmann formulation in PINN for weakly compressible flow problems.

A hybrid a posteriori MOOD limited lattice Boltzmann method to solve compressible fluid flows – LBMOOD

In this paper we blend two lattice-Boltzmann (LB) numerical schemes with an a posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solve hyperbolic systems of conservation laws in 1D and 2D. The first LB scheme is robust to the presence of shock waves but lacks accuracy on smooth flows. The second one has a second-order of accuracy but develops non-physical oscillations when solving steep gradients. The MOOD paradigm produces a hybrid LB scheme via smooth and positivity detectors allowing to gather the best properties of the two LB methods within one scheme. Indeed, the resulting scheme presents second order of accuracy on smooth solutions, essentially non-oscillatory behavior on irregular ones, and, an ‘almost fail-safe’ property concerning positivity issues. The numerical results on a set of sanity test cases and demanding ones are presented assessing the appropriate behavior of the hybrid LBMOOD scheme in 1D and 2D.

On the Limited Role of Conservation Laws in the Double Reduction Routine for Partial Differential Equations

The double reduction method for finding invariant solutions of a given partial differential equation (PDE) provides for the reduction of a $ q $-th order PDE that admits a nontrivial Lie symmetry and an associated nontrivial conservation law to an ordinary differential equation (ODE) of order $ q-1. $ In all the articles we have seen where the method has been used, the algorithm has involved writing the conservation law in canonical variables determined by the associated symmetry. In this paper, we illustrate that it is not necessary to use or even have the associated conservation law. It is enough to know that there exists a conservation law associated with a given Lie symmetry. Canonical variables derived from the symmetry are sufficient to achieve double reduction. In the canonical variables, the PDE is transformed after routine calculations into an ODE of order one less than that of the PDE. We have outlined steps involved in this variation of the double reduction method and illustrated the routine using five PDEs.

Dynamic Modeling of a HeXe-Cooled Mobile Nuclear Reactor with Closed Brayton Cycle

Helium-xenon (HeXe)-cooled mobile nuclear reactors have promising potential in future low-carbon energy systems. However, there is currently a lack of fast and reliable tools for analyzing the complicated dynamic characteristics of such systems. In this study, we developed a comprehensive dynamic modeling approach for a HeXe-cooled nuclear power system coupled with a closed Brayton cycle (CBC). The system’s key components, including the reactor, printed circuit heat exchanger (PCHE), and turbomachinery, are lumped-modeled to capture their time-varying behavior. A step-solving algorithm that incorporates HeXe mass conservation iteration is designed. The verification results demonstrate that the dynamic program is robust and reliable, with each time step converging within 25 iterations and the HeXe mass remaining within the range of 3.755 ± 0.01 kg throughout the simulation meeting the law of mass conservation. Then, a 1500 s frozen start-up simulation for the coupled system is conducted, in which the CBC is started in the first 500 s by increasing the main shaft speed to 40% of the rated value, and then the reactor is started by inserting external reactivity between 500 and 800 s. Both the dynamic process and the steady-state performance after the start-up are analyzed. The results show that the system achieved a stable electrical output of 5.7 MWe with a thermal efficiency of 32.5%. This study lays a solid foundation for future work aimed at improving the overall efficiency and performance of HeXe-cooled nuclear power systems.

Aspects of Developing an Innovative, Energy-Efficient, Low-Emission Co-Generator

Purpose. To develop a model and construct an experimental, innovative co-generator with high energy efficiency and low emissions. This involves developing a system that can efficiently generate both electricity and heat simultaneously while minimizing emissions, contributing to sustainability efforts, and addressing energy demands in a more environmentally friendly manner. Methodology. To achieve the goal, a system approach is utilized, enabling the selection of modelling types for the development of an experimental, cutting-edge co-generation system capable of efficiently producing both electricity and heat with superior energy efficiency and minimal emissions. For this purpose, the following steps were completed: processing and summarizing available literature and patent sources, analysing scientific and technical papers on the selection and application of modelling types in co-generation systems, and considering principles and individual approaches to input data formation for mathematical modelling. This process enables the selection of a mechanism and the creation of simulation models for effective energy production at various enterprises. Findings. Assessment is performed of the energy efficiency of the co-generator system under various operating conditions, comparing it with existing conventional methods of electricity and heat generation. Results are presented of performance testing to determine the system’s capability to simultaneously generate electricity and heat efficiently, considering factors such as output stability, load responsiveness, and overall reliability. Identification and evaluation are performed of innovative technologies and methodologies integrated into the co-generator design, highlighting their effectiveness in enhancing energy efficiency and reducing emissions. Insights into any operational challenges encountered during the construction, testing, and optimization phases, along with proposed solutions or improvements to solve these problems. The analysis of the overall environmental impact of deploying the co-generator showed potential benefits in terms of reduced greenhouse gas emissions and local air quality improvement. Originality. Using a combination of scientific approaches encompassing physics and heat transfer engineering, in accordance with the first law of thermodynamics, such as the conservation laws of energy and entropy, and principles of heat exchange employed to transfer heat between different mediums, gas kinetics have yielded values for the energy transformation coefficient, indicating qualitative characteristics of fuel thermolysis and power generation as the final product. Practical value. The results provide for developing a comprehensive model of an advanced co-generation system capable of efficiently producing both electricity and heat with superior energy efficiency and minimal emissions. It also entails determining the types of models for mathematical modelling at all management levels and establishing a new method for input data formation for both technologies and their subsystems, incorporating additional technological implementations.

Conservation law for electromagnetic fields in a space-time-varying medium and its implications

Conservation law for electromagnetic fields in a space-time-varying medium and its implications

Stochastic Lighthill-Whitham-Richards traffic flow model for nonlinear speed-density relationships

Stochasticity is becoming increasingly essential in traffic flow research, given its notable influence in several applications, such as real-time traffic management. To consider stochasticity in macroscopic traffic flow modeling, this paper introduces a stochastic Lighthill-Whitham-Richards (SLWR) model, which not only captures equilibrium values in steady-state conditions but also describes stochastic variabilities. The SLWR model follows a conservation law, in which the free-flow speed is randomized to represent heterogeneities of drivers. To more accurately reflect real-life traffic patterns, a nonlinear speed-density relationship is considered. For addressing this highly nonlinear problem, a dynamically bi-orthogonal (DyBO) method is coupled with the Taylor series expansion technique. The results of simulation experiments show that the SLWR model can effectively describe the evolution of stochastic dynamic traffic with temporal or geometric bottlenecks. Moreover, the DyBO solutions exhibit reasonable accuracy while significantly reducing computation costs compared with the Monte Carlo method.

Hybrid nanofluid flow in chamber containing heated and concentrated internal source under the effectiveness of applied magnetic field

Extensive applications of buoyancy-induced flow owing to temperature and concentration gradients have been found exclusively in diversified industrial and technological processes, such as heat exchange systems, solar collectors, heat sinks, and power plants. Additionally, installation of internal sources in confined configurations is essential in operation performance of many devices relevant to transformers, radiators, air cooling engines, fuel cells, semiconductor instruments and so forth. Furthermore, in recent decades, hybridized nanoparticles have been added to different setups to increase their efficiency. Therefore, the premier motive of current work is to examine mass and heat transport mechanisms in water-based fluids encompassed in a corrugated rectangular chamber with the induction of copper (Cu) and alumina (Al2O3) in a hybrid manner. A horizontally oriented magnetic field is assumed, along with the adjustment of a uniformly heated and concentrated source. The thermal conductivity relation proposed by Maxwell is obliged to scrutinize the performance of hybridized nanoparticles in controlling heat exchange in the domain. In this study, a power-law viscosity model is manifested along with utilization of other conservation laws and model structuring in the form of dimensionless expressions is performed by accomplishsing variables. Finite element simulations are executed by utilizing COMSOL Multiphysics 6.0 to solve developed governing equations system. The quantities of interest, such as average heat and mass fluxex, along with presentation of percentage change versus sundry factors, through graphical and tabular display. From a thorough examination, it is inferred that magnetic field plays an effective significance in controlling the excessive thermosolutal gradients. In addition, inudction of hybridized nanoparticles will positively affect thermal exchange in the base liquid compared to the situation when nanoparticles are not added. It is also deduced that average Nusselt and Sherwood numbers decreased up to 3 % and 2.4 % respectively for hydromagnetic situation (Ha = 0) in comparison to hydrodynamic case (Ha ≠ 0). Decrement in coefficients representing thermosolutal exchange reach up to 55 % and 66 % approximately against increment in number of corrugations. Induction of hybrid nanoparticles resulted in an increase in the coefficients of thermal and solutal transfer up to 14 % and 22 %, respectively, compared to the scenario where the base fluid contained no particles. For shear thinning aspects of non-Newtonain liquid, Nusselt and Sherwood numbers exceeds in comparison to Newtonian case and contrary aspects are observed for shear thickening materials.

An Improved Fifth-Order WENO Scheme for Solving Hyperbolic Conservation Laws Near Critical Points

An Improved Fifth-Order WENO Scheme for Solving Hyperbolic Conservation Laws Near Critical Points

A Local Macroscopic Conservative (LoMaC) Low Rank Tensor Method for the Vlasov Dynamics

In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum and energy at the discrete level. This is a follow-up work of our previous development of a conservative low rank tensor approach for Vlasov dynamics (arXiv:2201.10397). In that work, we applied a low rank tensor method with a conservative singular value decomposition to the high dimensional VP system to mitigate the curse of dimensionality, while maintaining the local conservation of mass and momentum. However, energy conservation is not guaranteed, which is a critical property to avoid unphysical plasma self-heating or cooling. The new ingredient in the LoMaC low rank tensor algorithm is that we simultaneously evolve the macroscopic conservation laws of mass, momentum and energy using a flux-difference form with kinetic flux vector splitting; then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables by a conservative orthogonal projection. The algorithm is extended to the high dimensional problems by hierarchical Tuck decomposition of solution tensors and a corresponding conservative projection algorithm. Extensive numerical tests on the VP system are showcased for the algorithm’s efficacy.

Random matrix approach to time-dependent forcing in many-body quantum systems

Changing some of its parameters over time is a paradigmatic way of driving an otherwise isolated many-body quantum system out of equilibrium, and a vital ingredient for building quantum computers and simulators. Here, we further develop a recently proposed nonlinear response theory which is based on typicality and random-matrix methods, and which is applicable to a wide variety of such parametrically perturbed systems in and out of equilibrium: We derive analytical approximations of the characteristic response function for the two limiting cases of fast driving and of strong and short-ranged-in-energy driving. Furthermore, we work out implications and predictions for common applications, including finite-time quenches and time-dependent forcing that breaks conservation laws of the underlying undriven system. Finally, we verify all predictions by numerical examples and discuss the theory's scope and limitations. Published by the American Physical Society 2024

Hydrodynamics of a discrete conservation law

AbstractThe Riemann problem for the discrete conservation law is classified using Whitham modulation theory, a quasi‐continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogs of well‐known dispersive hydrodynamic solutions—rarefaction waves (RWs) and dispersive shock waves (DSWs)—additional unsteady solution families and finite‐time blowup are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counterpropagating periodic wavetrains with the same frequency connected to a partial DSW or an RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations.

Lie symmetries, exact solutions and conservation laws of time fractional Boussinesq–Burgers system in ocean waves

In this paper, the Lie symmetry analysis method is applied to the time-fractional Boussinesq–Burgers system which is used to describe shallow water waves near an ocean coast or in a lake. We obtain all the Lie symmetries admitted by the system and use them to reduce the fractional partial differential equations with a Riemann–Liouville fractional derivative to some fractional ordinary differential equations with an Erdélyi–Kober fractional derivative, thereby getting some exact solutions of the reduced equations. For power series solutions, we prove their convergence and show the dynamic analysis of their truncated graphs. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.

Optimization of Legal Protection for the Leato Wreck Site: Discourse on Cultural Heritage Law and Maritime Conservation Law

Cultural heritage regulations and maritime conservation area regulations intersect and overlap in terms of legal protection, and nowhere is this more obvious than at the Leato underwater site/Japanese cargo ship wreck, referred to as the Leato wreck site in this paper. This legal issue later became a stumbling block in optimizing the legal protection of the Leato/Japanese cargo wreck underwater site. The study used two approaches: the statute approach and the conceptual approach. Analysis: a shipwreck can be defined as a catastrophic event that occurred in the past, with potential socio-economic benefits in the future, contingent upon optimal efforts to conserve history and culture, with great socio-economic potential if its exploitation is successfully optimized, with an eye to conservation efforts. The combination of these legally binding regulations is the key to optimizing the legal protection of the Leato underwater site. The maritime conservation area regulations safeguard the shipwreck's location as an integrated and interconnected ecosystem. By contrast, cultural heritage regulations protect the core zone of the Leato underwater site as a maritime cultural heritage from potential natural or human-caused damage. Other important issues that need to be considered are the empowerment of local communities in realizing a sustainable environment and cooperation between institutions in maintaining and preserving underwater cultural heritage.