Physical Equations Research Articles

An Unsupervised Inversion Method for Seismic Brittleness Parameters Driven by the Physical Equation

Brittleness is an important parameter characterizing the fracturing properties of shale reservoir, which can be predicted by the pre-stack seismic inversion. In order to overcome the low efficiency and ill-posed problems of the traditional pre-stack brittleness inversion, we propose a new unsupervised deep learning (DL) inversion method for seismic brittleness parameters based on the physical equation. This method integrates DL framework and the physical equation, and provides a DL inversion strategy without actual labels. We first input the original seismic data into the Fastformer network, and use the low-frequency model as the physical constraint to predict the brittle parameters. Then, the prediction results of brittleness parameters are sent to the forward modeling module (a linear approximation equation) to calculate the synthetic seismic data. Next, the error between the calculated seismic data and original seismic data is used to update the network prediction results. The network parameters are iteratively optimized to minimize the error, and the brittle prediction parameters are finally output. In the whole training process, it is not necessary to use the real brittle parameters as the labels. Through this method, the effect of approximate unsupervised learning is obtained. Finally, we apply the proposed method to the synthetic data and field data, and compared with the results inverted by the traditional L1 method. The experimental results show that the proposed method has higher inversion accuracy and efficiency than the traditional L1 method, which has a great potential in the practical application.

Clifford Algebras, Hypercomplex Numbers and Nonlinear Equations in Physics

Hypercomplex number systems are vector algebras with the definition of multiplication and division of vectors, satisfying the associativity and distributive law. In this paper, some new types of hypercomplex numbers and their fundamental properties are introduced, the Clifford algebra formalisms of hydrodynamics and gauge field equations are established, and some novel consistent conditions helpful to understand the properties of solutions to nonlinear physical equations are derived. The coordinate transformation and covariant derivatives of hypercomplex numbers are also discussed. The basis elements of the hypercomplex numbers have group-like properties and satisfy a structure equation $\A^2=n\A$. The hypercomplex number system integrates the advantages of algebra, geometry and analysis, and provides a unified, standard and elegant language and tool for scientific theories and engineering technology, so it is easy to learn and use. The description of mathematical, physical and engineering problems by hypercomplex numbers is of neat formalism, symmetric structure and standard derivation, which is especially suitable for the efficient processing of the higher dimensional complicated systems.

METHODS OF ACCOUNTING FOR TEMPERATURE AND STRAIN RATE VARIATION IN MULTILEVEL CONSTITUTIVE MODELS OF METAL DEFORMATION (ANALYTICAL REVIEW)

Technological metal forming processes involving hot and superplastic deformation are sensitive to temperature and strain rate. This is because inelasticity mechanisms operate in different ways under different conditions, leading to the formation of different structures and therefore different effective physical and mechanical properties of the material. Optimal temperature and strain rate conditions for the forming process which provide improved performance characteristics of the resulting products with acceptable energy consumption (or, conversely, minimum energy consumption with acceptable performance characteristics) can be most effectively determined by mathematical modeling. The key elements of the latter are constitutive models for describing the behavior of the material (physical equations), which can account for the influence of temperature and strain rate on various mechanisms of inelastic deformation. Such constitutive models can be most effectively developed using a multilevel approach based on the introduction of internal variables, crystal plasticity, and an explicit description of the material structure and physical deformation mechanisms. There are many works that propose multilevel mathematical models of metals that somehow explicitly account for the temperature and strain rate effects on inelastic deformation. Based on physical considerations, this analytical review defines the most promising approach to constructing multilevel constitutive models with comprehensive consideration of the temperature and strain rate effects.

Inverse design of microwave waveguide devices based on deep physics-informed neural networks

Using physics-informed neural networks to solve physical inverse problems is becoming a trend. However, it is difficult to solve the scheme that only introduces physical knowledge through the loss function. Constructing a reasonable loss function to make the results converge becomes a challenge. To address the challenge of physics-informed neural network models for inverse design of electromagnetic devices, a deep physics-informed neural network is introduced by using the mode matching method. The physical equations have been integrated into the network structure when the network is constructed. This feature makes the deep physics-informed neural network have a more concise loss function and higher computational efficiency when solving physical inverse problems. In addition, the training parameters of deep physics-informed neural networks are physically meaningful compared with those of traditional physics-informed neural networks. Users can control the network by parameters more easily. Taking the scattering parameter design of a two-port waveguide for example, we present a new metal topology inverse design scheme and give a detailed explanation. In numerical experiments, we target a set of physically realizable scattering parameters and inversely design the metallic septum by using a deep physics-informed neural network. The results show that the method can not only achieve the design target but also obtain solutions with different topologies. The establishment of multiple solutions is extremely valuable in implementing the inverse design. It can allow the designer to determine the size and location of the design area more freely while achieving the performance requirements. This scheme is expected to promote the application and development of the inverse design of electromagnetic devices.

A New General Solution of Axisymmetric Elastic Space Problem in Pavement Engineering

The general solution of the axial symmetry elastic space problem in cement concrete pavement is an elementary question. Combining the Love method and the Southwell method, the Southwell operator was used in the variable selection process, the displacement function was introduced to express the displacement component by the Love method, the expression of stress indicated by the displacement function was obtained by combining the displacement components, geometric equations, and physical equations, then the stress was substituted into the equilibrium equation, the biharmonic equation of displacement function was obtained by mathematical operation, and finally, a new general solution of the axisymmetric elastic space problem in cement concrete pavement was obtained. According to the proposed general solution, mechanical calculation for a cement concrete pavement slab on Winkler foundation was carried out, the displacement curve at the top surface of the slab and the stress curve at the top, middle, and bottom surface of the slab were derived, and the results showed that the method is feasible in the calculation of cement concrete pavement. The general solution provided a new method for solving the axisymmetric elastic space problem of cement concrete pavement.

Fluid-structure interaction and band gap analysis of periodic composite liquid-filled pipe

This paper studies the vibration band gap characteristics of a liquid-filled composite pipeline. The transfer matrix method is used to create a one-dimensional fluid–structure interaction model of the composite pipeline from the constitutive equation, physical equations, and boundary conditions of the anisotropic material. A phononic crystal pipeline is designed with a periodic arrangement on the axial pipe wall. The method used to analyze the fluid–structure interaction of liquid-filled composite and phononic crystal pipelines is verified by comparing responses with finite-element method results. The results’ agreement demonstrates the method’s and calculation code’s correctness. Furthermore, the band gaps for an empty composite phononic crystal pipe, water hammer, and a liquid-filled pipe are calculated by Bloch vector theory, and the effects of fluid–structure interaction and the composite parameters on the three band gap characteristics are analyzed. The numerical results show that Poisson’s coupling affects the band gap at some frequency points, while the laying angle and fiber volume fraction in the composite pipe influence the band gap length and amplitude change. The research results of this paper provide a reference for the design and vibration control of liquid-filled composite pipelines.

CONSTRUCTION OF FRACTAL SOLITON SOLUTIONS FOR THE FRACTIONAL EVOLUTION EQUATIONS WITH CONFORMABLE DERIVATIVE

In this paper, the fractional evolutions are described by using the conformable derivative for the first time. We implement fractional functional variable method (FFVM) to obtain some new kinds of fractal soliton wave solutions for these fractional evolution equations. The simplicity and effectiveness of this proposed method are tested on the fractional Drinfeld–Sokolov system and fractional cubic Klein–Gordon equation. The FFVM provides a new perspective to construct exact fractal soliton wave solutions of complex fractional nonlinear evolution equations in mathematical physics.

The balance equation in quantum physics

The balance equation in quantum physics

Non-differentiable fractional odd-soliton solutions of local fractional generalized Broer-Kaup system by extending Darboux transformation

In this paper, a local fractional generalized Broer-Kaup (gBK) system is first de?rived from the linear matrix problem equipped with local space and time fractional partial derivatives, i.e, fractional Lax pair. Based on the derived fractional Lax pair, the second kind of fractional Darboux transformation (DT) mapping the old potentials of the local fractional gBK system into new ones is then established. Finally, non-differentiable frcational odd-soliton solutions of the local fractional gBK system are obtained by using two basic solutions of the derived fractional Lax pair and the established fractional DT. This paper shows that the DT can be extended to construct non-differentiable fractional soliton solutions of some local fractional non-linear evolution equations in mathematical physics.

EXPLICIT SOLUTIONS FOR THE CONFORMABLE REGULARIZED LONG WAVE BURGER'S EQUATION

In this paper, a dynamical analysis of the conformable regularized long-wave burgers equation is carried out with help of improved $ \tan\big(\frac{\phi(\eta)}{2}\big) $-expansion method. Fractional complex transform converts a nonlinear fractional differential equation in an ordinary differential form which resulted into a number of exact solutions like exponential function solutions, hyperbolic function solutions, trigonometric function solutions and rational function solutions. The constarint conditions are also given for each solution. The physical profiles of proposed solutions are portrayed by 3D and 2D graphs as well as the influence of fractional parameter is also studied for some solutions. Our proposed results showed that improved $ \tan\big(\frac{\phi(\eta)}{2}\big) $-expansion method is reliable method to solve the nonlinear equation in mathematical physics.

An application of spectral linearisation method on the steady two-dimensional radial flow of Au-water and Ag-water nanofluids between two inclined plane walls

The present work purveys the steady two-dimensional viscous incompressible radial flow of Au-water and Ag-water nanofluids between plane walls which either converge or diverge in the presence of MHD effect. A uniform magnetic field is applied. The governing partial differential equations of the present physics and their appropriate boundary conditions are initially cast into the dimensionless forms to reduce into the ordinary differential equation. The resulting equation thus formed is then solved by adopting the spectral linearisation method (SLM) to get the accurate numerical solution. Solution errors and residual norms were analysed to elaborate the convergence and accuracy of the numerical solution. The present results are validated with favourable comparisons to previously published results due to the current investigations' unique cases. Parametric study of the governing parameters, namely the magnetic field strength parameter M, Reynolds number Re, angle of inclination α, and the volume fraction parameter ɸ on the non-dimensional velocity is conducted. The analysis discloses that the profiles of the flow are largely impacted by the physical parameters.

NEW SOLITARY WAVE SOLUTIONS FOR THE FRACTIONAL JAULENT–MIODEK HIERARCHY MODEL

The main goal of this paper is to study the new solitary wave behaviors of the fractional Jaulent–Miodek hierarchy model (FJMHE) with M-truncated fractional derivative. First, we use the fractional sech-function method (FSFM) to obtain some new solitary wave solutions of the fractional Jaulent–Miodek hierarchy equation. The new method is simple and effective, which provides a more powerful mathematical technique for exploring solitary wave solutions of the fractional evolution equations in mathematical physics. Finally, some 3D and 2D graphs are employed to illustrate the physical properties of the obtained new solitary wave solutions.

Electron Shape Calculated for the Dual-Charge Dual-Mass Model

A model for the internal structure of the electron using classical physics equations has been previously published by the author. The model employs both positive and negative charges and positive and negative masses. The internal attributes of the electron structure were calculated for both ring and spherical shapes. Further examination of the model reveals an instability for the ring shape. The spherical shape appears to be stable, but relies on tensile or compressive forces of the electron material for stability. The model is modified in this document to eliminate the dependency on material forces. Uniform stability is provided solely by balancing electrical and centrifugal forces. This stability is achieved by slightly elongating the sphere along the spin axis to create a prolate ellipsoid. The semi-major axis of the ellipsoid is the spin axis of the electron, and is calculated to be 1.20% longer than the semi-minor axis, which is the radius of the equator. Although the shape deviates slightly from a perfect sphere, the electric dipole moment is zero. In the author’s previously published document, the attributes of the internal components of the electron, such as charge and mass, were calculated and expressed as ratios to the classically measured values for the composite electron. It is interesting to note that all of these ratios are nearly the same as the inverse of the Fine Structure Constant, with differences of less than 15%. The electron model assumed that the outer surface charge was fixed and uniform. By allowing the charge to be mobile and the shape to have a particular ellipticity, it is shown that the calculated charge and mass ratios for the model can be exactly equal to the Fine Structure Constant and the Constant plus one. The electron radius predicted by the model is 15% greater than the Classical Electron Radius.

Investigation on aortic hemodynamics based on physics-informed neural network.

Pressure in arteries is difficult to measure non-invasively. Although computational fluid dynamics (CFD) provides high-precision numerical solutions according to the basic physical equations of fluid mechanics, it relies on precise boundary conditions and complex preprocessing, which limits its real-time application. Machine learning algorithms have wide applications in hemodynamic research due to their powerful learning ability and fast calculation speed. Therefore, we proposed a novel method for pressure estimation based on physics-informed neural network (PINN). An ideal aortic arch model was established according to the geometric parameters from human aorta, and we performed CFD simulation with two-way fluid-solid coupling. The simulation results, including the space-time coordinates, the velocity and pressure field, were obtained as the dataset for the training and validation of PINN. Nondimensional Navier-Stokes equations and continuity equation were employed for the loss function of PINN, to calculate the velocity and relative pressure field. Post-processing was proposed to fit the absolute pressure of the aorta according to the linear relationship between relative pressure, elastic modulus and displacement of the vessel wall. Additionally, we explored the sensitivity of the PINN to the vascular elasticity, blood viscosity and blood velocity. The velocity and pressure field predicted by PINN yielded good consistency with the simulated values. In the interested region of the aorta, the relative errors of maximum and average absolute pressure were 7.33% and 5.71%, respectively. The relative pressure field was found most sensitive to blood velocity, followed by blood viscosity and vascular elasticity. This study has proposed a method for intra-vascular pressure estimation, which has potential significance in the diagnosis of cardiovascular diseases.

Physical Knowledge-Enhanced Deep Neural Network for Sea Surface Temperature Prediction

Traditionally, numerical models have been deployed in oceanography studies to simulate ocean dynamics by representing physical equations. However, many factors pertaining to ocean dynamics seem to be ill-defined. We argue that transferring physical knowledge from observed data could further improve the accuracy of numerical models when predicting Sea Surface Temperature (SST). Recently, the advances in earth observation technologies have yielded a monumental growth of data. Consequently, it is imperative to explore ways in which to improve and supplement numerical models utilizing the ever-increasing amounts of historical observational data. To this end, we introduce a method for SST prediction that transfers physical knowledge from historical observations to numerical models. Specifically, we use a combination of an encoder and a generative adversarial network (GAN) to capture physical knowledge from the observed data. The numerical model data is then fed into the pre-trained model to generate physics-enhanced data, which can then be used for SST prediction. Experimental results demonstrate that the proposed method considerably enhances SST prediction performance when compared to several state-of-the-art baselines.

Physics-Informed Sparse Neural Network for Permanent Magnet Eddy Current Device Modeling and Analysis

The objective of this letter is to study the prediction of the electromagnetic field and the output performance of permanent magnet eddy current devices based on a physics-informed sparse neural network (PISNN). In order to achieve this goal, a unified physical model is firstly defined according to different types of permanent magnet eddy current devices, which is equivalent to solving a parameterized magnetic quasi-static problem. A soft constraint module and a hard constraint module, composed of physical equations, are constructed. The soft constraints are then integrated into the neural network's objective function, while the hard constraint module is utilized to predict device performance and physical field. Stochastic gradient descent is used to minimize the residual of the physical equations during PISNN training. Subsequently, the structural parameters and operating parameters of PMECD are modified to verify the generalization ability of the model. Our results indicate that PISNN accurately and efficiently predicts the electromagnetic field distribution and the output torque. Furthermore, our prediction results for permanent magnet eddy current devices with different parameters demonstrate the potential of the proposed method for transfer learning.

Design and Modeling of Multi-vertebrae Continuum Robot for Minimally Invasive Surgery

The constant curvature assumption is normally utilized in kinematic modeling of continuum manipulators due to its simplicity. Nevertheless, in practical applications, factors such as frictional disturbances may cause the curvature of a continuum manipulator to deviate from being constant. In this article, we design a novel multi-vertebrae continuum manipulator that utilizes the uniform joint deformation curvature of a superelastic NITI alloy rod. We propose kinematic modeling approach for a multi-vertebrae continuum manipulator. To improve the unevenness of joint angles and increase motion accuracy, we inserted elastic beams into the multi-vertebrae continuum manipulator. The design takes into account frictional forces, and we established geometric constraint equations and force equilibrium equations for the hinged continuum manipulator. We employed an identification method to establish the physical equations of the deformation section. By solving these equations and analyzing the geometric relationships, we obtained the mapping relationship between the driving screw force and the end displacement of the continuum manipulator. To prove the effectiveness of the proposed method, we performed MATLAB simulations. This paper serves as a reference for the design and modeling of future minimally invasive surgical manipulators.

Variable Physical Constants and Beyond

We previously revealed that the speed of light in vacuum c, the gravitational constant G, the vacuum permittivity ε, and the vacuum permeability μ can be defined by the temperature T (or the expected average frequency f) of cosmic microwave background (CMB) radiation. Given that CMB is continuously cooling, that is, T is continuously decreasing, we proposed that the above “constants” are variable and their values at some space-time with CMB temperature T (cT, GT, εT, and μT) can be described using their values (c0, G0, ε0, and μ0) and the temperature (T0) of CMB at present space-time. Based on the above observation, a number of physical equations related with these constants are re-described in this study, including relativity equation, mass-energy equation, and Maxwell’s equations, etc.

Diverse solitary wave solutions of fractional order Hirota-Satsuma coupled KdV system using two expansion methods

The generalized Hirota-Satsuma coupled KdV system with fractional-order derivative plays a significant role to simulate the interaction of nearby identical-weight particles in a crystal lattice structure, two long waves interaction with different dispersion relationships, the description of shallow-water wave propagation, ion-acoustic waves, plasma physics and some other fields. In this study, diverse analytical wave solutions in general and standard structures are established of the stated equation by introducing two viable techniques, namely, the generalized Kudryashov scheme and the two variables G'/G,1/G-expansion approach. The solutions are established in terms of elementary functions, specifically trigonometric functions, rational functions, exponential functions, and hyperbolic functions. For definite values of free parameters, the obtained analytical wave solutions transform into solitary wave solutions. The graphical patterns of the wave solutions with there-, two-, and contour plots are depicted magnificently to elucidate the internal structure of the phenomenon. The methods contribute as powerful mathematical tools and appear to be further effective, computerized, and user-friendly to investigate nonlinear fractional equations as well as comprehensive analytical solutions of nonlinear evolution equations in engineering, technology, and mathematical physics.

The Analysis of Experiment Video on Cognitive Conflict-Based Teaching Materials to Enhance Momentum-Impulse Concepts Understanding

Video analysis experiments can be used to construct physics concepts and equations. Integrating video analysis experiments into cognitive conflict learning allows students to increase their understanding of concepts. Some research results show that students need clarification about momentum and impulse material, low conceptual understanding, and misconceptions occur. The problem of misconceptions can hinder student learning progress. One solution to overcome these problems is the application of physics teaching materials based on cognitive conflict integrated with real experiment video analysis. This study aims to analyze the effectiveness of these teaching materials in improving students’ understanding of the concept of momentum and impulse. This study used a quasi-experimental method with a nonequivalent control group design. The sample consisted of 72 high school students with two sample classes, the experimental and the control classes, which were selected using the cluster random sampling technique. The instrument in this study was a concept test that was valid and reliable. Data were analyzed using Mann-Whitney test nonparametric statistics with the help of IBM SPSS Statistic 25. The results obtained by Sig. Asim < 0.05, in the rejection area of H0, means that teaching materials influence students’ understanding of concepts on momentum and impulse. Thus, conflict-based teaching materials integrating real experiment video analysis have been effective in increasing students’ conceptual understanding.