Physical Equations Research Articles

Implementation of the HHL Algorithm for Solving the Poisson Equation on Quantum Simulators

The Poisson equation is a fundamental equation of mathematical physics that describes the potential distribution in static fields. Solving the Poisson equation on a grid is computationally intensive and can be challenging for large grids. In recent years, quantum computing has emerged as a potential approach to solving the Poisson equation more efficiently. This article uses quantum algorithms, particularly the Harrow–Hassidim–Lloyd (HHL) algorithm, to solve the 2D Poisson equation. This algorithm can solve systems of equations faster than classical algorithms when the matrix A is sparse. The main idea is to use a quantum algorithm to transform the state vector encoding the solution of a system of equations into a superposition of states corresponding to the significant components of this solution. This superposition is measured to obtain the solution of the system of equations. The article also presents the materials and methods used to solve the Poisson equation using the HHL algorithm and provides a quantum circuit diagram. The results demonstrate the low error rate of the quantum algorithm when solving the Poisson equation.

Magnetic field and dissipation effects on mixed convection viscous fluid flow by a channel in the presence of porous medium and heat generation/absorption phenomenon

AbstractIn this study, we investigate the phenomenon of mixed convection in viscous fluid flow by a vertical channel, considering the presence of magneto‐hydro‐dynamics and porosity. Mixed convection occurs when both buoyancy forces and external forces, such as pumps or fans, influence the flow behavior. Understanding and accurately predicting mixed convection is crucial for optimizing heat exchanger design and performance. To model the temperature equation, we utilize the concept of the first law of thermodynamics. Additionally, we incorporate effects such as Joule heating, heat generation, and radiative heat flux in the energy equation modeling. The resulting physical liquid equations are solved using the shooting technique with the RKF (Runge–Kutta–Fehlberg) scheme. We present graphical representations of the flow variables, discussing their behavior in detail. The introduction provides an overview of the paper's roadmap, while the conclusion highlights the main and significant results obtained from our study. We found that, the temperature is more when liquid motion in between channel for large numerical values of thermal radiation parameter and Reynolds number.

Active learning modules for a numerical methods course

Education research has suggested that students who receive a wider variety of sensory inputs during the learning process will be able to make more connections and more deeply understand course content. To this end, three physical hands-on laboratory learning modules were designed for a computational numerical methods course to help students develop intuition for physical systems, to allow for a comparison of real data against numerical solutions of physics equations, and to increase student engagement. These modules were designed to reinforce specific topics covered in the numerical methods curriculum, including numerical differentiation and numerical solution of ordinary differential equations. Pre- and post-tests showed an increase in student knowledge after module completion. In addition, end-of-term surveys showed that a strong majority of students believed modules helped them visualize and understand key numerical methods concepts, and that most would be interested in learning new material with similar modules in the future.

Integral Method for Determining the Stress State of an Anisotropic Reinforced Concrete Shell

The article deals with the issue of estimating the strength and stress-strain state of a reinforced concrete shell by the method of membrane theory and the theory of infinitesimal bending. A brief description of the methodology consists in the fact that the ground stress state of the shell is allocated to an independent problem, when, without introducing boundary effects, two of the four boundary conditions of the general moment theory are distinguished, which, together with the equations of the momentless theory, determine the ground stress state, and then boundary effects are superimposed. The equilibrium equations of the moment theory in forces and moments, geometric equations, components of tangential deformation and displacement, and the physical equations of state connecting them, expressing forces and moments through the components of deformation, are presented, since the characteristics of the stress-strain state of the shell depend not only on the variability of external influences and forces, but also on the length of the structure. The provisions laid down in the article shall be retained in the case of casings made of anisotropic material, provided that the presented elastic ratio is met. A system of equations of the generalized semi-membrane (semi-bending) state of an arbitrary shell of zero curvature, which determines the accuracy of this approach, is solved. The equations of the semi-torque theory for a circular cylindrical shell are presented, as well as describing the semi-membrane stress state of a long shell of zero curvature. The most important result of the study is the method of constructing integrals of the ground stress state of the shell based on the method of simple iterations, which expands the possibilities of the reliability theory, which makes it possible to build the foundations for the practical calculation of reinforced concrete shells according to the membrane theory and the theory of infinitesimal bending.

On the study of bright, dark and optical wave structures for the coupled fractional nonlinear Schrödinger equations in plasma physics

On the study of bright, dark and optical wave structures for the coupled fractional nonlinear Schrödinger equations in plasma physics

Physical Equations Relating Extract and Relative Density

Equations for converting between relative density and percent by mass extract based upon physically meaningful values (rather than fitted parameters) are derived from physical principles. An exact equation and two approximate formulas using one and two physically meaningful parameters respectively are given based on the partial specific volumes of sucrose and water in solution, which are calculated for sucrose concentrations varying between 0 and 20% by mass fraction. The predictions of these equations are compared to the American Society of Brewing Chemists (ASBC) tables and the accuracy of the predicted values is analyzed. The root mean squared error over the entire range of the ASBC tables of the approximate formula with two parameters is less than 0.016°P and that of the formula using only one parameter is less than 0.023°P. The approximate formula with one parameter provides a simple and easily remembered, but nevertheless highly exact means for any brewer to calculate extract without specialized analytical devices.

Enhancing Friction Models for Starting Up Water Installations Containing Trapped Air

Starting up water installations is typically a task that falls within the purview of water utility companies. These operations involve the presence of two separate fluids (water and air) that can be analyzed in terms of consideration two distinct behaviors (hydraulic and thermodynamic). During a filling process, trapped air pockets exhibit a trend of declining volume, generating pressure surges that are typically not addressed under current worldwide regulations. This research introduces an innovative mathematical approach based on physical equations to investigate filling operations in water installations involving trapped air, incorporating an unsteady friction model (using the Brunone friction coefficient), in combination with the rigid water column model. The validation of the proposed model is carried out in an experimental facility measuring 7.36 m in length. The proposed model is then applied to a case study involving a 460 m long pipeline with an internal pipe diameter of 150 mm, featuring an undulating profile composed of three branches, to demonstrate how the gravity term should be calculated in real-world water installations. The results showed that the proposed model, considering an unsteady friction model, is suitable for simulating the start up of water pipelines for the experimental facility analysis and the case study. The Swamee–Jain formula yielded the best results compared to other formulations for computing the friction factor.

Applying Physics-Informed Neural Networks to Solve Navier–Stokes Equations for Laminar Flow around a Particle

In recent years, Physics-Informed Neural Networks (PINNs) have drawn great interest among researchers as a tool to solve computational physics problems. Unlike conventional neural networks, which are black-box models that “blindly” establish a correlation between input and output variables using a large quantity of labeled data, PINNs directly embed physical laws (primarily partial differential equations) within the loss function of neural networks. By minimizing the loss function, this approach allows the output variables to automatically satisfy physical equations without the need for labeled data. The Navier–Stokes equation is one of the most classic governing equations in thermal fluid engineering. This study constructs a PINN to solve the Navier–Stokes equations for a 2D incompressible laminar flow problem. Flows passing around a 2D circular particle are chosen as the benchmark case, and an elliptical particle is also examined to enrich the research. The velocity and pressure fields are predicted by the PINNs, and the results are compared with those derived from Computational Fluid Dynamics (CFD). Additionally, the particle drag force coefficient is calculated to quantify the discrepancy in the results of the PINNs as compared to CFD outcomes. The drag coefficient maintained an error within 10% across all test scenarios.

Modeling of additive manufacturing processes with time‐dependent material properties using physics‐informed neural networks

AbstractRecently, physics‐informed neural networks (PINNs) have been effectively utilized in a wide range of problems within the domains of applied mathematics and engineering. In PINNs, the governing physical equations are directly incorporated into the loss function of the network and a conventional labeled dataset is not required for its training. In order to successfully simulate the additive manufacturing processes with concrete, a novel process‐based FE‐simulation has been developed where the Drucker–Prager plasticity model is used as the material model. In this work, we will examine the deployment of a PINN to substitute the Newton–Raphson iterations that occur in the return‐mapping algorithm of the Drucker–Prager plasticity model.

Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised pochhammer-chree equation in mathematical physics

Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised pochhammer-chree equation in mathematical physics

Combination of Physics-Informed Neural Networks and Single-Relaxation-Time Lattice Boltzmann Method for Solving Inverse Problems in Fluid Mechanics

Physics-Informed Neural Networks (PINNs) improve the efficiency of data utilization by combining physical principles with neural network algorithms and thus ensure that their predictions are consistent and stable with the physical laws. PINNs open up a new approach to address inverse problems in fluid mechanics. Based on the single-relaxation-time lattice Boltzmann method (SRT-LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator, the PINN-SRT-LBM model is proposed in this paper for solving the inverse problem in fluid mechanics. The PINN-SRT-LBM model consists of three components. The first component involves a deep neural network that predicts equilibrium control equations in different discrete velocity directions within the SRT-LBM. The second component employs another deep neural network to predict non-equilibrium control equations, enabling the inference of the fluid’s non-equilibrium characteristics. The third component, a physics-informed function, translates the outputs of the first two networks into physical information. By minimizing the residuals of the physical partial differential equations (PDEs), the physics-informed function infers relevant macroscopic quantities of the flow. The model evolves two sub-models that are applicable to different dimensions, named the PINN-SRT-LBM-I and PINN-SRT-LBM-II models according to the construction of the physics-informed function. The innovation of this work is the introduction of SRT-LBM and discrete velocity models as physical drivers into a neural network through the interpretation function. Therefore, the PINN-SRT-LBM allows a given neural network to handle inverse problems of various dimensions and focus on problem-specific solving. Our experimental results confirm the accurate prediction by this model of flow information at different Reynolds numbers within the computational domain. Relying on the PINN-SRT-LBM models, inverse problems in fluid mechanics can be solved efficiently.

A new method for detecting dark Matter and explaining formulas which describe fluid matter

Nowadays there is a new challenge among scientists, all of us know dark energy spreads the universe by constant index, but the big question is whether the universe expanding without stopping. I am not agreeing, because on the other side, we have dark matter and we know the force of dark matter and dark energy are opposite of each other. In this article, it has been tried to explain how to detect dark fluid and made an effort to find a formula that describe the relationship between dark matter and dark energy or totally dark fluid. Throughout the history, many scientists had the idea to estimate the behavior of dark matter by the means of time, it means they desired to see what is the effect of time on the movement of dark fluid and influence of it on expanding and diminishing physical universe. Cosmology, attached to the theory of dark matter and dark energy, elaborates that both of them are in a single framework and they are not separated into ptomaine. In the last section I added time to predict dark fluid treatment by passing time. To pursue this goal, at first an experiment was designed in which by assist of cigarette smoke and electrical light, the behavior of dark fluid was observed by the author. Then, by means of quantum physic equations and adding time as a stable factor to quantum physic equation, a model and formula was coined by which it was possible to show dark fluid movement. It is very significant to state that this final formula was not completely accomplished due to the fact the gravity of black hole was not regarded in this experiment. Asan important factor, by help of cosmic noise, voyager program of NASA, information theory is related to lost galaxy information.

ARCHITECTURES OF NEURAL NETWORKS BASED ON THE PHYSICAL CHARACTERISTICS OF THE OBJECT

Neural networks based on the physical characteristics of an object have been actively used in recent time in order to solve combined problems that require multi-step calculations. Therefore this technology shall be thoroughly studied and properly used. Neural networks solve many problems in different fields of science. Today there are several different ways of application, one of the possible applications is the solution of the equation of partial variables, in special cases the approximation of piecewise functional systems can be used. This article presents an overview of neural network architecture based on physical characteristics of the object. The application of physical informative neural networks (PINNs) for approximation of piecewise functional systems is considered. It is shown that PINNs allow to use information about the physical equations of the system in training, automatically determine boundaries between ranges of input variables, and use various activation functions and hidden layers.

Physics symbolic learner for discovering ground‐motion models via NGA‐West2 database

AbstractGround‐motion model (GMM) is the basis of many earthquake engineering studies and practices. In this study, a novel physics‐informed symbolic learner (PISL) method is proposed to automatically discover mathematical equation operators as symbols. The sequential threshold ridge regression algorithm is utilized to distill a concise and interpretable explicit characterization of complex systems of ground‐motions. In addition to the basic variables retrieved from previous GMMs, the current PISL incorporates three priori physical conditions, namely, distance, magnitude, and site amplification saturation. Based on the Next Generation Attenuation West2 database, GMMs developed using the PISL, an empirical regression method (ERM), and an artificial neural network (ANN) are compared in terms of residuals and extrapolation of peak ground acceleration and velocity. The results show that the inter‐ and intra‐event standard deviations of the three methods are similar. The functional form of the PISL is more concise than that of the ERM and ANN. The extrapolation capability of the PISL is more accurate than that of the ANN. The PISL‐GMM used in this study provide a new paradigm of regression that considers both physical‐ and data‐driven machine learning and can be used to identify the implied physical relationships and prediction equations of ground‐motion variables.

Two-temperature generalized magneto-thermic-viscoelastic response problem considering non-local and memory effects

Based on the two-temperature generalized magneto-thermal-viscoelastic theory considering scale effects and fractional time derivatives, the dynamic response of generalized magneto-thermal-viscoelastic theory in homogeneous isotropic semi-infinite elastic media with infinite conductivity is studied. Laplace Fourier integral transform and numerical inverse transform are used to solve the general solution of dimensionless physical equations. Based on the numerical results, the effects of time factor, fractional parameter and scale effect on the transient magneto thermic viscoelastic response of the structure are discussed respectively. The above results are visually expressed in the form of graphs. The results show that the temperature and displacement are less affected by each parameter, the stress increases with the increase of time, non-local and viscosity parameters, and decreases with the increase of fractional order parameters. This result is of great significance to the application of different physics fields, and provides some theoretical guidance for solving practical engineering problems.

Viscoelastic rod using the generalized finite difference method

The finite difference method is quite extensively used to obtain the approximate solutions of many equations of mathematical physics. In this study, the precise algorithm in the time domain is combined with the generalized finite difference method to solve dynamic viscoelasticity problems. The numerical results obtained are satisfactory, and they are presented together with finite difference and finite element solutions.

Formulating a Novel Physical Equation to Quantify Interocular Visual Acuity Influence

ABSTRACT The limitation of conventional visual acuity assessment, which primarily focuses on individual eye performance (monocular visual acuity tests). This study addresses this limitation by emphasizing the importance of binocular vision, where both eyes work together. Binocular vision provides numerous advantages, such as improved depth perception, a wider field of view, reduced visual fatigue, and binocular summation. The article introduces a new mathematical model based on the physics of constructive interference in light waves and the anatomy of visual nerves. This model yields an equation that quantifies how weakness in one eye affects the visual acuity of the other. By analyzing the correlation between the rates of visual weakness in each eye and the corresponding distances, utilizing the Snellen 6-meter test, valuable insights are gained into the complex interplay between eye deficiencies and their collective impact on overall visual acuity. The equations derived from this study can have significant implications for eye care professionals, clinicians, and patients, contributing to a more comprehensive understanding of visual impairments and enriching the field of optics and vision science.

Deep and machine learning architecture for high accuracy prediction of interior ballistics for modular charges

AbstractInterior ballistics modelling intends to calculate physical quantities by solving equations requiring a set of input like thermodynamical properties linked to propellant or mechanical properties like friction coefficients, linked to the system, that are often difficult to assess experimentally. Using a consequent and high‐quality dataset of thousands of 155 mm modular charge experiments, we investigated how deep and machine learning approaches could build an efficient model for predicting muzzle velocity, pressure, and ignition time. After identifying the main variables influencing the physics and several potentially relevant to the machine learning and deep learning algorithms, a stacking ensemble learning model (SEL) was built. It is composed of eight machine learning algorithms and one deep learning algorithm, all as weak learner, reaching a consensus for predicting experimental outputs through a strong learner. The performance assessment, on a 900 experiments test data set that were not used in training, shows a prediction accuracy of 97 % at 1 % accuracy for muzzle velocity, 93 % at 3 % for pressure and 72 % at 5 ms for ignition time. The prediction ability for muzzle velocity within 2 m/s reaches 73 % that can be considered as sufficient. Furthermore, the SEL model even having zero knowledge about physics can discriminate gun barrel erosion effect at high and low pressure. This approach allows to imagine a way of modelling physical phenomena poorly described by physical equations or requiring unmeasurable figures.

Optical soliton solutions for time-fractional Ginzburg–Landau equation by a modified sub-equation method

In the present work, we employed a novel modification of the Sardar sub-equation approach, leading to the successful derivation of several exact solutions for the time-fractional Ginzburg–Landau equation with Kerr law nonlinearity. These solutions encompass a range of categories, including singular, wave, bright, mixed dark–bright, and bell-shaped optical solutions. We demonstrate the dynamic behavior and physical significance of these optical solutions of the proposed model via several graphical simulations, including contour plots, three-dimensional (3D) graphs, and two-dimensional (2D) plots. Furthermore, we investigate the magnitude of the time-fractional Ginzburg–Landau equation by analyzing the influence of the conformable fractional order derivative and the impact of the time parameter on the newly constructed optical solutions. The proposed technique is a generalized form that incorporates various methods, including the improved Sardar sub-equation method, the modified Kudryashov method, the tanh-function extension method, and others. To the best of our knowledge, these solutions are novel and have not been reported in the literature. Moreover, the present method is efficient and robust for analyzing applied differential equations in plasma physics and nonlinear optics.

Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis

The efficient representation of random fields on geometrically complex domains is crucial for Bayesian modelling in engineering and machine learning, including Gaussian process regression and statistical finite element analysis. Today’s prevalent random field representations are either intended for unbounded domains or are too restrictive in terms of possible field properties. Because of these limitations, new techniques leveraging the historically established link between stochastic PDEs (SPDEs) and random fields have been gaining interest in the statistics and engineering literature. The SPDE representation is especially appealing for engineering applications with complex geometries which already have a finite element discretisation for solving the physical conservation equations. In contrast to the dense covariance matrix of a random field, its inverse, the precision matrix, is usually sparse and equal to the stiffness matrix of an elliptic SPDE. In this paper, we use the SPDE representation to develop a scalable framework for large-scale statistical finite element analysis and Gaussian process (GP) regression on geometrically complex domains. The statistical finite element method (statFEM) introduced by Girolami et al. (2022) is a novel approach for synthesising measurement data and finite element models. In both statFEM and GP regression, we use the SPDE formulation to obtain the relevant prior probability densities with a sparse precision matrix. The properties of the priors are governed by the parameters and possibly fractional order of the SPDE so that we can model on bounded domains and manifolds anisotropic, non-stationary random fields with arbitrary smoothness. We use for assembling the sparse precision matrix the same finite element mesh used for solving the physical conservation equations. The observation models for statFEM and GP regression are such that the posterior probability densities are Gaussians with a closed-form mean and precision. The expressions for the mean vector and the precision matrix do not contain dense matrices and can be evaluated using only sparse matrix operations. We demonstrate the versatility of the proposed framework and its convergence properties with one and two-dimensional Poisson and thin-shell examples.