Physical Equations Research Articles

THERMAL PROCESSES OF FIRE DETECTORS’ HEAT-RESISTANT SENSITIVE ELEMENTS

The purpose of the article. The article deals with thermal processes in the heat-resistant sensitive element of fire detectors when an electric current flows through it. Such processes are described by an inhomogeneous equation of mathematical physics with boundary conditions of the third kind. Methodology. The Hankel integral transformation is used to solve the differential equation that describes the thermal processes in the heat-resistant sensitive element of fire detectors. As a result of the transformations, a general expression was obtained for the temperature of the heat-resistant sensitive element of fire detectors under the condition that an electric current flows through it without restrictions on its parameters. This expression is represented as a series of Bessel functions. Results. The results of scientific research confirmed that due to the small size of the fire detectors’ sensitive element, the expression for its temperature is presented in the form of an average value over the sensitive element volume. Scientific novelty. For the first time, a general expression for the temperature of the heat-resistant sensitive element of fire detectors is obtained, which allows obtaining its main technical characteristics, and also serves as a basis for moving to other mathematical descriptions. Practical value. Also, using the integral Laplace transformation, an expression for the transfer function of the fire detectors’ heat-resistant sensitive element was obtained and it was confirmed that with an error of no more than 4,6 %, it is described by the transfer function of the inertial link.

Blow-up criteria of the simplified Ericksen–Leslie system

In this paper, we establish scaling invariant blow-up criteria for a classical solution to the simplified Ericksen–Leslie system in terms of the positive part of the second eigenvalue of the strain matrix and orientation field in mixed-norm Lebesgue spaces. Our result may be also regarded as an extension or improvement of the corresponding results obatined by Neustupa and Penel (Trends in Partial Differential Equations of Mathematical Physics, pp. 197–212, 2005), Miller (Arch. Ration. Mech. Anal. 235(1):99–139, 2020) and Huang and Wang (Commun. Partial Differ. Equ. 37(5):875–884, 2012).

Learning-Based Approaches to Current Identification from Magnetic Sensors.

Direct measurement of electric currents can be prevented by poor accessibility or prohibitive technical conditions. In such cases, magnetic sensors can be used to measure the field in regions adjacent to the sources, and the measured data then can be used to estimate source currents. Unfortunately, this is classified as an Electromagnetic Inverse Problem (EIP), and data from sensors must be cautiously treated to obtain meaningful current measurements. The usual approach requires using suited regularization schemes. On the other hand, behavioral approaches are recently spreading for this class of problems. The reconstructed model is not obliged to follow the physics equations, and this implies approximations which must be accurately controlled, especially if aiming to reconstruct an inverse model from examples. In this paper, a systematic study of the role of different learning parameters (or rules) on the (re-)construction of an EIP model is proposed, in comparison with more assessed regularization techniques. Attention is particularly devoted to linear EIPs, and in this class, a benchmark problem is used to illustrate in practice the results. It is shown that, by applying classical regularization methods and analogous correcting actions in behavioral models, similar results can be obtained. Both classical methodologies and neural approaches are described and compared in the paper.

Hyd1d and hydflood - R packages to compute water levels and annual flood durations along the rivers Rhine and Elbe, Germany

Floodplains are environments characterized by fluctuating water levels. Spatial and temporal heterogeneity of habitat conditions driven by hydrology are the key factor for their high biodiversity because strong environmental filtering either creates favorable conditions for specialized organisms or eliminates competing generalists.hyd1d and hydflood are two R packages for simplified hydrological modelling in two large central European floodplains – along the rivers Rhine and Elbe in Germany – that enable users to simulate the dynamics of water level changes, flood extents and durations with high spatial and temporal resolution over long time periods and vast areas. They are simplified in the sense of not being based on physical equations, but they implement fast computing interpolation of 1d hydraulic data and GIS extrapolation algorithms. Annual and long-term products provide the possibility to predict the distribution of floodplain habitats.All input datasets are highly integrated into the packages and automatically updated. Computationally expensive products such as annual flood durations between 1960 and 2022 are provided as open data. The software itself is provided as open source software through CRAN and in public code repositories. With minimal efforts users can apply the software within the given conditions or integrate other river systems through data and source code contributions.

Use of high-speed angiography HSA-derived boundary conditions and Physics Informed Neural Networks (PINNs) for comprehensive estimation of neurovascular hemodynamics.

Physics-informed neural networks (PINNs) and computational fluid dynamics (CFD) have both demonstrated an ability to derive accurate hemodynamics if boundary conditions (BCs) are known. Unfortunately, patient-specific BCs are often unknown, and assumptions based upon previous investigations are used instead. High speed angiography (HSA) may allow extraction of these BCs due to the high temporal fidelity of the modality. We propose to investigate whether PINNs using convection and Navier-Stokes equations with BCs derived from HSA data may allow for extraction of accurate hemodynamics in the vasculature. Imaging data generated from in vitro 1000 fps HSA, as well as simulated 1000 fps angiograms generated using CFD were utilized for this study. Calculations were performed on a 3D lattice comprised of 2D projections temporally stacked over the angiographic sequence. A PINN based on an objective function comprised of the Navier-Stokes equation, the convection equation, and angiography-based BCs was used for estimation of velocity, pressure and contrast flow at every point in the lattice. Imaging-based PINNs show an ability to capture such hemodynamic phenomena as vortices in aneurysms and regions of rapid transience, such as outlet vessel blood flow within a carotid artery bifurcation phantom. These networks work best with small solution spaces and high temporal resolution of the input angiographic data, meaning HSA image sequences represent an ideal medium for such solution spaces. The study shows the feasibility of obtaining patient-specific velocity and pressure fields using an assumption-free data driven approach based purely on governing physical equations and imaging data.

Identification of swirling air flow velocity by non-neutrally buoyant tracer particle based on machine learning

In the non-intrusive measurement of swirling air flow, helium-filled soap bubbles (HFSBs) are ideal neutrally buoyant tracer particles However, there are some researchers that do not use HFSBs in the non-intrusive measurement of swirling air flow, leading to some kind of measurement inaccuracy. Since the flow velocity data has been implicitly included in the physical equations of any kind of tracer particles, it is possible to extract such hidden flow velocity from particle trajectory. In this study we propose a physics-informed procedure of adopting SINDy algorithm to identify the hidden physical equations of non-neutrally buoyant particle dynamics, so that the implicit flow velocity can be discovered. First of all, the numerical experiment is conducted to generate particle trajectory in a 2D swirling air flow in small cyclone separator. Based on the numerical experiment trajectory data, the input variables for SINDy algorithm are properly constructed. The output of SINDy algorithm, which are the identified physical equations, are evaluated and validated on two different-density particle trajectory data. Our results show that the physical equations of tracer particle dynamics can be identified and the discovered flow velocity data has a maximum deviation of 1.4% from the truth (R2≥0.999). The proposed method may remove the requirement of NB tracer particle in non-intrusive measurement of swirling air flow, and may be applied to recognize the physical equations of complex particle laden flow.

A General Paradigm for Retrieving Soil Moisture and Surface Temperature from Passive Microwave Remote Sensing Data Based on Artificial Intelligence

Soil moisture (SM) and land surface temperature (LST) are entangled, and the retrieval of one of them requires a priori specification of the other one. Due to insufficient observational information, retrieval of LST and SM from passive microwave remote sensing data is often ill-posed, and the retrieval accuracy needs to be improved. In this study, a novel fully-coupled paradigm is developed to robustly retrieve SM and LST from passive microwave data, which integrates deep learning, physical methods, and statistical methods. The key condition of the general paradigm proposed by us is that the output parameters of deep learning can be uniquely determined by the input parameters theoretically through a certain mathematical equation. Firstly, the physical method is deduced based on the energy radiation balance equation. The nine unknowns require the brightness temperatures of nine channels to construct nine equations, and the solutions of the physical method equations are obtained by model simulation. Based on the derivation of the physical method, the solution of the statistical method is constructed using multi-source data. Secondly, the solutions of physical and statistical methods constitute the training and test data of deep learning, which is used to obtain the solution curve of physical and statistical methods. The retrieval accuracy of LST and SM is greatly improved by smartly utilizing the mutual prior knowledge of SM and LST and cross iterative optimization calculations. Finally, validation indicates that the mean absolute error of the retrieved SM and LST data are 0.027 m3/m3 and 1.38 K, respectively, at an incidence angle of 0–65°. A model-data-knowledge-driven and deep learning method can overcome the shortcomings of traditional methods and provide a paradigm for retrieval of other geophysical variables. The proposed paradigm not only has physical meaning, but also makes deep learning physically interpretable, which is a milestone in the retrieval of geophysical remote sensing parameters based on artificial intelligence technology.

Application of Newton–Raphson Method for Computing the Final Air–Water Interface Location in a Pipe Water Filling

The estimation of thermodynamic behavior during filling processes with entrapped air in water pipelines is a complex task as it requires solving a system of algebraic-differential equations. A lot of different numerical methods have been used for this purpose in literature including the rigid water column (RWC) model. The main advantage of the RWC model is its acceptable accuracy with very low computational load. In that context, this research presents the computation of critical points of the physical equations that describe the phenomenon. These points provide information about the final position of the air–water interface. The Newton–Raphson method was then applied to obtain a unique equation that can be used by engineers to directly compute variables such as air pocket pressure and water column length at the end of the hydraulic event. A case study was analyzed to compare the results of the mathematical model with the obtained equation for computing critical points. Both methods provided the same values for the water column length at the end of the hydraulic event. A sensitivity analysis was conducted to identify dependent and non-dependent parameters for evaluating the critical points. The proposed formulation was validated through an experimental set of data.

Stress Distribution in Direct Shear Loading and its Implication for Engineering Failure Analysis

Shear stress concentrations may promote damage and failure processes. Quantities of studies have focused on the direct shear loading test, while the analytical model has not yet been studied in depth. Aiming to fill the knowledge gap, the theoretical and numerical analyses of the shear stress distribution in the shear band were investigated. In order to reflect the variation in the stress state, the differential element method was first used. The shear stress distribution equation was derived from the stress equilibrium, geometric and physical equations. The shear stress distribution was plotted, using the proposed equation. After that, the ratio of yield strength to crack initiation strength was calculated. The analytical model was analyzed with FDEM simulation, and the results were compared with those obtained from the laboratory tests. Using the elastoplastic theory, the damage evolution and process in rock were characterized from laboratory scale. The implication for underground engineering analysis was finally discussed with a case study of strain rockburst in hard rock. The analytical model and results could provide a fundamental basis for stability analysis in geotechnical engineering.

The Good, the Bad and the ‘Entropic Potential of an Event’

The definitions of “good” and “bad” typically belong to the field of ethics. Does physics have a property that can distinguish and group “good” and “bad” events using only its own physical instruments and equations? Although the physical property “entropy” as a measure of chaos appears to be the prime candidate for such a grouping, it is in fact unsuitable for a detailed analysis. However, the “Entropic Potential of an Event”, which describes the influence of the current event to the future change in entropy, perfectly suits this role. While the second law of thermodynamics dictates the direction of entropy change in an isolated system it does not dictate the speed of entropy growth. The speed of entropy growth on Earth is changing in particular due to human-related events, which ethics calls "good" or "bad". Such events, which ethics defines as "good", always eventually decelerate entropy growth. In contrast, events which ethics defines as "bad", always eventually accelerate entropy growth. This article presents the methods of calculating the “Entropic Potential of an Event” for the cases: “A commander receives an order to bombard a city”, “A cancer tumor is growing inside a human’s body” and other. This article also stresses the importance of the “Time factor” since only for a sufficiently large time interval T the “En-tropic Potential of an Event” Z(T, A) can be estimated and potentially precisely calculated. This article also checks on a scale of several centuries if real-life events are averaged in such a way that their entropic potentials become negligible. This analysis shows that prior 1750 the averaged entropic potential of events occurred in human society was negative and had the value of approximately 10^17 bits (value is an initial approximation). The situation after the industrial revolution is more complicated, as past 1750 human civilization began to utilize non-renewable resources (oil, coal, gas, etc.) for warming, cooling, and transportation with a corresponding entropy growth. The term “entropy” is applicable to a very wide range of events, from physics and chemistry to art and information. Correspondingly, the significant advantages of the “Entropic Potential of an Event” as a physical foundation of the intuitive terms “good” and “bad” is its measurability, ability to compare events of entirely different natures and its universality Weak point of "Utilitarianism" is inability to quantify, compare, or measure happiness or well-being which is postulated to be the ultimate goal of moral behavior. The "Entropic Potential of an Event" solves this problem and presents in formulas (and sometime calculates in bits) the Entropic Potentials of events we normally label as "good" and "bad".

Development of air purifier operation guidelines using grey box models for the concentrations of particulate matter in elementary school classrooms.

Considering the hazardous effects of particulate matter (PM) exposure on students and teachers and the high PM concentration issue in South Korea, air purifiers have recently been installed in most classrooms to improve air quality. However, some on-site challenges, such as operational costs and noise, have been issues with the continuous operation of air purifiers. Therefore, a guideline is needed to dynamically predict the indoor PM concentration based on the changes in outdoor PM concentration and activate the air purifiers only when necessary. This study develops a grey-box model that uses measured data and physical differential equations to perform the given objective and verifies its accuracy using ASTM D5157. Modeling and analysis results have obtained information that can form the basis for developing guidelines to address PM issues in schools: The air purifier should be operated during periods where the predicted values exceed the limit in closed windows and the air purifier is not operating. It was also confirmed that the need for the operation of the air purifier varies between schools and classrooms under the same outdoor PM concentration. Indoor PM concentration increased significantly after students' simultaneous mass movement, necessitating air purifiers' operation before and after the events. The prefilter of the heater also aided in the removal of coarse PM. Additionally, the limitations and future development directions of the model were discussed.

Reference shape of bodies with finite incompatible deformations

The work develops differential-geometric methods for modeling of finite incompatible deformations of hyperelastic solids. Deformation incompatibility can be caused, for example, by inhomogeneous temperature fields and distributed defects. As a result, residual stresses and distortion of geometric shape of the body occur. These factors determine the critical parameters of modern high-precision technologies, in particular, in additive manufacturing technologies. In this regard, the development of methods for their quantitative description is an urgent problem of modern solid mechanics.
 The application of methods of differential geometry is based on the representation of a body as a smooth manifold equipped with a metric and a non-Euclidean connection. This approach allows one to interpret the body as a global stress-free shape and to formulate the physical response and material balance equations with respect to this shape. Within the framework of the geometric method, deformations are characterized by embeddings of non-Euclidean shape into physical space, which is still considered to be Euclidean. Measures of incompatibility are identified with the invariants of the affine connection, namely, torsion, curvature, and nonmetricity, and the connection itself is determined by the type of physical process.

POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier–Stokes equations

We present a Reduced Order Model (ROM) which exploits recent developments in Physics Informed Neural Networks (PINNs) for solving inverse problems for the Navier–Stokes equations (NSE). In the proposed approach, the presence of simulated data for the fluid dynamics fields is assumed. A POD-Galerkin ROM is then constructed by applying POD on the snapshots matrices of the fluid fields and performing a Galerkin projection of the NSE (or the modified equations in case of turbulence modeling) onto the POD reduced basis. A POD-Galerkin PINN ROM is then derived by introducing deep neural networks which approximate the reduced outputs with the input being time and/or parameters of the model. The neural networks incorporate the physical equations (the POD-Galerkin reduced equations) into their structure as part of the loss function. Using this approach, the reduced model is able to approximate unknown parameters such as physical constants or the boundary conditions. A demonstration of the applicability of the proposed ROM is illustrated by three cases which are the steady flow around a backward step, the flow around a circular cylinder and the unsteady turbulent flow around a surface mounted cubic obstacle.

Physics informed neural networks for electric field distribution characteristics analysis

Electric field calculations based on the Poisson equation have been widely used in high voltage and plasma technology. However, in practical applications, the electric field distribution in space is relatively complex, and the simulation technology based on the traditional method is often a simplification of reality, which leads to a large error between the simulation and the actual measured value. In the actual application process, due to the limitation of measurement methods, it is necessary to infer the electric field data at other locations in space according to the measurement results. Physics informed neural networks (PINNs) are introduced into the electric field calculation. PINNs are considered partial differential equation solvers based on deep neural networks. In this paper, 2D and 3D electric field distributions are discussed and compared with the finite element method. A method of dividing the dielectric distribution based on the sigmoid function is proposed, which can be effectively used to construct the spatial electric field model of the homogeneous dielectric. The combination of the data and physical model based on PINNs establishes a method to solve the inverse problem of the relative permittivity in the electric field. The results show that PINNs can calculate the distribution of the electric field according to the physical equations and different types of constraints and parameters.

The Influence of Deformation Phase-Space on Spectra of Heavy Quarkonia in Improved Energy Potential at Finite Temperature Model of Shrodinger Equation Via the Generalized Boob’s Shift Method and Standard Perturbation Theory

In this work, we obtain solutions of the deformed Schrödinger equation (DSE) with improved internal energy potential at a finite temperature model in a 3-dimensional nonrelativistic noncommutative phase-space (3D-NRNCPS) symmetries framework, using the generalized Bopp’s shift method in the case of perturbed nonrelativistic quantum chromodynamics (pNRQCD). The modified bound state energy spectra are obtained for the heavy quarkonium system such as charmonium cc- and bottomonium bb- at finite temperature. It is found that the perturbative solutions of the discrete spectrum are sensible to the discreet atomic quantum numbers (j,l,s,m) of the ( QQ- (Q=c,b)) state, the parameters of internal energy potential (T,αs(T), mD (T),β,c), which are the Debye screening mass mD (T), the running coupling constant αs(T) the critical temperature β, the free parameter c in addition to noncommutativity parameters (Θ,θ-). The new Hamiltonian operator in 3D-NRNCPS symmetries is composed of the corresponding operator in commutative phase-space and three additive parts for spin-orbit interaction, the new magnetic interaction, and the rotational Fermi-term. The obtained energy eigenvalues are applied to obtain the mass spectra of heavy quarkonium systems (cc- and bb-). The total complete degeneracy of the new energy levels of the improved internal energy potential changed to become equal to the new value 3n2 in 3D-NRNCPS symmetries instead of the value n2 in the symmetries of 3D-NRQM. Our non-relativistic results obtained from DSE will possibly be compared with the Dirac equation in high-energy physics.

Birational solutions to the set-theoretical 4-simplex equation

The 4-simplex equation is a higher-dimensional analogue of Zamolodchikov’s tetrahedron equation and the Yang–Baxter equation which are two of the most fundamental equations of mathematical physics. In this paper, we introduce a method for constructing 4-simplex maps, namely solutions to the set-theoretical 4-simplex equation, using Lax matrix refactorisation problems. Employing this method, we construct 4-simplex maps which at a certain limit give tetrahedron maps classified by Kashaev, Korepanov and Sergeev. Moreover, we construct a Kadomtsev–Petviashvili type of 4-simplex map. Finally, we introduce a method for constructing 4-simplex maps which can be restricted on level sets to parametric 4-simplex maps using Darboux transformations of integrable PDEs. We construct a nonlinear Schrödinger type parametric 4-simplex map which is the first parametric 4-simplex map in the literature.

Design and Implementation of an Einsteinian Energy Learning Module

The most famous equation in physics, E = mc2, is rarely introduced in middle school physics curricula. Recent research has shown that teaching Einsteinian concepts at the middle school level is feasible and beneficial. This paper analyses an Einsteinian energy teaching module for Year 8 students (13–14 years old), which encompasses the two fundamental energy formulas in modern physics, E = mc2 and E = hf. In the context of activity-based learning, the Einsteinian energy module relates to all the forms of energy in traditional school curricula. This study uses a design-based research approach within the Model of Educational Reconstruction framework. Modern experiments, historical events, and educational research helped us identify relevant Einsteinian energy concepts, activities, and assessments. The study included 22 students who participated in nine in-class Einsteinian energy lessons. Analysing results in the post-test showed a 31% mean increase from the pre-test, a clear and significant positive change in students’ conceptual understanding. The results demonstrated students’ ability to deal with very large and small constants of proportionality and physical concepts involved in the module.

Scattering from infinity for semilinear wave equations satisfying the null condition or the weak null condition

We show global existence backward from scattering data at infinity for semilinear wave equations satisfying the null condition or the weak null condition. Semilinear terms satisfying the weak null condition appear in many equations in physics. The scattering data is given in terms of the radiation field, although in the case of the weak null condition there is an additional logarithmic term in the asymptotic behavior that has to be taken into account. Our results are sharp in the sense that the solution has the same spatial decay as the radiation field does along null infinity, which is assumed to decay at a rate that is consistent with the forward problem. The proof uses a higher order asymptotic expansion together with a new fractional Morawetz estimate with strong weights at infinity.

Towards modelling of energy storages for use in an intelligent energy management system

AbstractWith the transition towards renewable energy underway, demand‐side management together with the local generation of renewable energy is receiving growing attention. Optimizing the self‐consumption of locally produced renewable energy can not only have financial benefits for the respective household or business (and improve their autarky from increasingly unstable energy markets), but also help improve grid stability against the volatility of some sources of renewable energy. In order to optimize the self‐consumption of a given household or enterprise with an energy management system, the energy demand and generation, as well as the behaviour of all controlled devices have to be forecast. There are many different methods of mathematically modelling these different properties, but the task becomes especially challenging for dynamic systems like energy storage systems. Here, the state of charge at a specific time point depends on both external influences and the previous time point's state of charge, thus modelling errors quickly accumulate. In the context of small and medium agricultural enterprises, examples of such dynamic systems can be electrical storages, like a battery, or thermal storages like a milk cooling tank or a heat pump.In this work, a common way for the modelling of these systems is explored, namely parameter identification. Here, the parameters of an ordinary differential equation representing the assumed physical behaviour of the system are identified from measured data of states and controls in an optimization problem. The success of this approach, and with it the performance of the derived model depends on how well the physical equations actually describe the system, but also on the quality, quantity and content of the measurement data used. In the case of a household battery storage already a simple physical model with parameters identified from measurement data of state of charge and battery power provides useful results. The example of a milk cooling tank provides a situation where external forcing has an important influence on the system's state of charge, its temperature. The cleaning of the tank generates a large heat influx, and raises the temperature, while the filling and emptying of milk influences the heat capacity. While these influences are shown to also be representable by equations based on the physical processes, parameter identification becomes difficult when they are not included in the measurement data. Thus, different approaches to derive these external forcings from the available data are outlined.

‘Openness in Action’ Early Steps in Cosmic Phenomenology

‘Openness in Action’ Early Steps in Cosmic Phenomenology