Equations Of Mechanics Research Articles

On Some Results of the Nonuniqueness of Solutions Obtained by the Feynman–Kac Formula

The Feynman–Kac formula establishes a link between parabolic partial differential equations and stochastic processes in the context of the Schrödinger equation in quantum mechanics. Specifically, the formula provides a solution to the partial differential equation, expressed as an expectation value for Brownian motion. This paper demonstrates that the Feynman–Kac formula does not produce a unique solution but instead carries infinitely many solutions to the corresponding partial differential equation.

Acerca de la relación entre la mecánica cuántica y las finanzas

The present article has as a goal to show an alternative way for deriving the Black-Scholes equation using the Schrödinger equation of quantum mechanics, taking into account that both of these partial differential equations are quite similar.

An intelligent identification method based on self-adaptive mechanism regulated neural network for chemical process

BackgroundThe complex characteristics of chemical process, such as multivariable, nonlinear, time-varying and strong coupling often lead to the poor effect of traditional identification theory in practical application. The development of deep learning in recent years has brought a breakthrough for nonlinear system identification, but more progress is still needed. MethodsThis paper proposes a nonlinear identification method based on self-adaptive mechanism regulated Long Short-Term Memory (LSTM) network for chemical process dynamic simulation. First, to increase the reliability of the application of neural network (NN) for chemical process identification and improve the generalization ability, the known differential equation describing the mechanism is taken as a regularization term to constrain the output of the NN. Then, a specific training method is proposed, which introduces trainable self-adaptive weights to force the neural network to focus on the regions with large training error. In addition, a semi-supervised network training method is proposed for the case that some parameters in the mechanism equation are unknown. Finally, a dynamic virtual device (VD) model is established, which can simulate the dynamic response of controlled objects. Significant findingsTo evaluate the efficiency of the developed identification method, various comparative experiments are conducted on pH neutralization and continuous stirred tank reactor (CSTR) processes. The experimental results show that the proposed identification method can obtain a nonlinear dynamic model with robustness, high accuracy and strong generalization ability.

Power Dissipation and Wear Modeling in Wheel–Rail Contact

This paper is concerned with the modeling of power dissipation due to friction and its relation with wear estimation in wheel–rail contact. Wear is a complex multi-scale and multi-physical phenomenon appearing in rolling contact. Wear is generated by high contact stress and the work of friction forces. This phenomenon leads to the appearing of the worn material in the form of wear debris between contacting surfaces. In contact models, wear is usually described in terms of the wear depth function. This function modifies the gap between the contacting bodies as well as the shape of the surfaces of the wheel and rail in contact. In this paper, besides the wear depth function, the dissipated energy, rather than the contact stress, is taken into account to evaluate the wear impact on rail or wheel surfaces. The dissipated energy allows us to more precisely evaluate the wear debris amount as well as the depth of wear and its distribution along the contact interface. A two-dimensional rolling contact problem with frictional heat flow is considered. The elasto-plastic deformation of the rail is considered. This contact problem is governed by a coupled system of mechanical and thermal equations in terms of generalized stresses, displacement and temperature. The finite element method is used to discretize this problem. A discretized system of equations with nonpenetration and friction conditions is transformed and formulated as a nonlinear complementarity problem. The generalized Newton method is applied to numerically solve this mechanical subproblem. The Cholesky method is used to find the solution of the heat-conductive problem. The dissipated power is evaluated based on the resultant force and slip at a reference point. Numerical results including the distribution of slip velocity, power factor and wear rate are provided and discussed.

A fast algorithm for the Schrödinger equation in quaternionic quantum mechanics

The eigenvalue problem of a Hermitian quaternion matrix plays a crucial role in quaternion quantum mechanics because it is closely related to the solution of Schrödinger equation. In this paper, a fast algorithm is proposed for finding the eigenvalues and corresponding eigenvectors of a Hermitian quaternion matrix based on the real representation of a quaternion matrix as well as the special structure and properties of a Hermitian quaternion matrix. Numerical experiments demonstrate that, compared with the existing computational methods for the eigenvalue problem of a Hermitian quaternion matrix, the proposed method in this paper not only greatly improves the computational efficiency, but also achieves better experimental results in terms of the corresponding computational errors.

The effect of inspiratory rise time on mechanical power calculations in pressure control ventilation: dynamic approach

BackgroundMechanical power may serve as a valuable parameter for predicting ventilation-induced injury in mechanically ventilated patients. Over time, several equations have been developed to calculate power in both volume control ventilation (VCV) and pressure control ventilation (PCV). Among these equations, the linear model mechanical power equation (MPLM) closely approximates the reference method when applied in PCV. The dynamic mechanical power equation (MPdyn) computes power by utilizing the ventilatory work of breathing parameter (WOBv), which is automatically measured by the mechanical ventilator. In our study, conducted in patients with Covid-19 Acute Respiratory Distress Syndrome (C-ARDS), we calculated mechanical power using both the MPLM and MPdyn equations, employing different inspiratory rise times (Tslope) at intervals of 5%, ranging from 5 to 20% and compared the obtained results.ResultsIn our analysis, we used univariate linear regression at both I:E ratios of 1:2 and 1:1, considering all Tslope values. These analyses revealed that the MPdyn and MPLM equations exhibited strong correlations, with R2 values exceeding 0.96. Furthermore, our Bland–Altman analysis, which compared the power values derived from the MPdyn and MPLM equations for patient averages and all measurements, revealed a mean difference of −0.42 ± 0.41 J/min (equivalent to 2.6% ± 2.3%, p < 0.0001) and −0.39 ± 0.57 J/min (equivalent to 3.6% ± 3.5%, p < 0.0001), respectively. While there was a statistically significant difference between the equations in both absolute value and relative proportion, this difference was not considered clinically relevant. Additionally, we observed that each 5% increase in Tslope time corresponded to a decrease in mechanical power values by approximately 1 J/min.ConclusionsThe differences between mechanical power values calculated using the MPdyn and MPLM equations at various Tslope durations were determined to lack clinical significance. Consequently, for practical and continuous mechanical power estimation in Pressure-Controlled Ventilation (PCV) mode, the MPdyn equation presents itself as a viable option. It is important to note that as Tslope times increased, the calculated mechanical power exhibited a clinically relevant decrease.

Complex structure-preserving method for Schrödinger equations in quaternionic quantum mechanics

Complex structure-preserving method for Schrödinger equations in quaternionic quantum mechanics

SIMULATION OF ELASTOPLASTIC FRACTURE OF A CENTER CRACKED PLATE

The strength of a square plate with a central crack at normal separation was studied within the framework of the Neuber–Novozhilov approach using a modified Leonov–Panasyuk–Dugdale model using an additional parameter, the diameter of the plasticity zone (width of the pre-fracture zone). As a model of a deformable solid body, a model of an ideal elastic-plastic material with a limiting relative elongation was chosen. This class of materials includes, for example, low-alloy steels used in structures operating at temperatures below the cold brittleness threshold. In the presence of a singular feature in the stress field in the vicinity of the crack tip, it is proposed to use a two-parameter discrete integral strength criterion. The deformation fracture criterion is formulated at the tip of a real crack, and the force criterion for normal stresses, taking into account averaging, is formulated at the tip of a model crack. The lengths of real and model cracks differ by the length of the pre-fracture zone. The constitutive equations of the analytical model are analyzed in detail depending on the characteristic linear dimension of the material structure. Simple formulas suitable for verification calculations for the critical breaking load and the length of the pre-fracture zone are obtained. Numerical modeling of the propagation of plasticity zones in square plates under quasi-static loading has been performed. In the numerical model, the updated Lagrangian formulation of the equations of mechanics of a deformable solid body is used, which is most preferable for modeling the deformation of bodies made of an elastic-plastic material at large deformation. The plastic zone in the vicinity of the crack tip is obtained by the finite element method. The results of analytical and numerical prediction of plate fracture under plane deformation are compared. It is shown that the results of numerical experiments are in good agreement with the results of calculations using the analytical model of fracture of materials with a structure under normal separation. Diagrams of quasi-brittle and quasi-ductile fracture of a structured plate are constructed.

The Nanoscale Wetting Parameter and Its Role in Interfacial Phenomena: Phase Transitions in Nanopores.

Through analysis of the statistical mechanical equations for a thin adsorbed film (gas, liquid, or solid) on a solid substrate or confined within a pore, it is possible to express the equilibrium thermodynamic properties of the film as a function of just two dimensionless parameters: a nanoscale wetting parameter, αw, and pore width, H*. The wetting parameter, αw, is defined in terms of molecular parameters for the adsorbed film and substrate and so is applicable at the nanoscale and for films of any phase. The main assumptions in the treatment are that (a) the substrate structure is not significantly affected by the adsorbed layer and (b) the diameter of the adsorbate molecules is not very small compared to the spacing of atoms in the solid substrate. We show that different surface geometries of the substrate (e.g., slit, cylindrical, and spherical pores) and various models of wall heterogeneity can be accounted for through a well-defined correction to the wetting parameter; no new dimensionless variables are introduced. Experimental measurements are reported for contact angles for various liquids on several planar substrates and are shown to be closely correlated with the nanoscale wetting parameter. We apply this approach to phase separation in nanopores of various geometries. Molecular simulation results for the phase diagram in confinement, obtained by the flat histogram Monte Carlo method, are reported and are shown to be closely similar to experimental results for capillary condensation, melting, and the triple point. The value of the wetting parameter, αw, is shown to determine the qualitative behavior (e.g., increase vs decrease in the melting temperature, capillary condensation vs evaporation), whereas the pore width determines the magnitude of the confinement effect. The triple point temperature and pressure for the confined phase are always lower than those for the bulk phase for all cases studied.

Lagrange multiplier and variational equations in mechanics

Lagrange multiplier and variational equations in mechanics

Approximate deconvolution discretisation

A new strategy is presented for the construction of high-order spatial discretisations extracted from a lower-order basic discretisation. The key consideration is that any spatial discretisation of a derivative of a solution can be expressed as the exact differentiation of a corresponding ‘filtered’ solution. Hence, each numerical discretisation method may be directly linked to a unique spatial filter, expressing the truncation error of the basic method. By approximately deconvolving the implied filter of the basic numerical discretisation an augmented high-order method can be obtained. In fact, adopting a deconvolution of the implied filter of suitable higher order enables the formulation of a new spatial discretisation method of correspondingly higher order. This construction is illustrated for finite difference (FD) discretisation schemes, solving partial differential equations in fluid mechanics. Knowing the implied filter of the basic discretisation, one can derive a corresponding higher order method by approximately eliminating the implied spatial filter to a certain desired order. We use deconvolution to compensate for the implied filter. This corresponds to a ‘sharpening’ of numerical solution features before the application of the basic FD method. The combination will be referred to as Approximate Deconvolution Discretisation (ADD). The accuracy of the deconvolved FD scheme depends on the order of approximation of the deconvolution filter. We present the ‘sharpening’ of several well-known FD operators for first- and second-order derivatives and quantify the achieved accuracy in terms of the modified wavenumber spectrum. Examples include high-order extensions up to new schemes with spectral accuracy. The practicality of the deconvolved FD schemes is illustrated in various ways: (i) by investigation of exactly solvable advection and diffusion problems, (ii) by tracking the evolution of the numerical solution to the Taylor-Green vortex problem and (iii) by showing that ADD yields spectral accuracy for the Burgers equation and for double-jet flow of an incompressible fluid.

Effects of Different Flotation Agents on the Nucleation and Growth of Potassium Chloride

The flotation agent is an important collector in the production of potassium chloride and is brought into the crystallization stage with the reflux of the mother liquor. Octadecylamine Hydrochloride (ODA), 1-Dodecylamine Hydrochloride (DAH) and Sodium 1-dodecanesulfonate (SDS) were selected to study their effect on the nucleation of potassium chloride. Focused Beam Reflectance Measurement was used to collect the nucleation-induced periods of KCl in the presence of flotation agents at different supersaturations. Then, empirical equations, classical nucleation theory and growth mechanism equations were employed for data analysis. It was found that the presence of flotation agents increased the nucleation sequence m, and m(ODA) &gt; m(SDS) &gt; m(DAH) &gt; m(H2O). In addition, the interfacial energy data obtained using classical nucleation theory suggest that the flotation agents used in our paper promoted the homogeneous nucleation of KCl (reduced from 5.3934 mJ·m−2 to 5.1434 mJ·m−2) and inhibited the heterogeneous nucleation of KCl (increased from 2.8054 mJ·m−2 to 3.6004 mJ·m−2). This investigation also revealed that the growth of potassium chloride was consistent with the 2D nucleation-mediated growth mechanism, and the addition of flotation agent did not change the growth mechanism of potassium chloride. Finally, the particle size distribution results were exactly consistent with the order of nucleation order m. The study of nucleation kinetics and growth mechanisms of different flotation agents on potassium chloride can provide guidance for optimizing the production process of potassium chloride and developing new flotation agents.

The continuity equation in the phase space quantum mechanics

A quantum phase space version of the continuity equation for systems with internal degrees of freedom is derived. The 1-D Dirac equation is introduced and its phase space counterpart is found. The phase space representation of free motion and of scattering in a nonrelativistic and relativistic case for setups with internal degrees of freedom is discussed and illustrated. Properties of Wigner functions of unbound states are analysed.

Novel soliton molecule solutions for the second extend (3+1)-dimensional Jimbo-Miwa equation in fluid mechanics

The main aim of this paper is to investigate the different types of soliton molecule solutions of the second extend (3+1)-dimensional Jimbo-Miwa equation in a fluid. Four different localized waves: line solitons, breather waves, lump solutions and resonance Y-type solutions are obtained by the Hirota bilinear method directly. Furthermore, the molecule solutions consisting of only line waves, breathers or lump waves are generated by combining velocity resonance condition and long wave limit method. Also, the molecule solutions such as line-breather molecule, lump-line molecule, lump-breather molecule, etc. consisting of different waves are derived. Meanwhile, higher-order molecule solutions composed of only line waves are acquired.

MAGNETOHYDRODYNAMICS AND POWER-LAW FLUIDS IN DOUBLE LID-DRIVEN CAVITY WITH SEMI-CIRCULAR BODIES

The use of complex fluids is one of the modern techniques used in small devices in order to enhance their thermal performance. This paper is a numerical study of a complex fluid imprisoned in a chamber with two hot bodies exposed to a magnetic field of constant intensity. The upper and lower walls move horizontally at a constant velocity, while the lateral sides are thermally insulated. The numerical simulation of the system was achieved based on the finite volume method that solves the differential equations of fluid mechanics and heat transfer. Simulations were carried out under the following conditions: Re = 1 to 40, Ri = 0 to 100, n = 0.6 to 1.4 and Ha = 0 to 100. The study showed that the thermal activity of the two bodies is different and related to initial condition. Also, the effect of the magnetic field is strong in the case of shear-thinning fluids, while its effect is diminished in the case of shear-thickening fluids.

Determination of the main technological parameters of cyclic-steam stimulation of oil reservoirs with account of heat losses along the borehole

Due to the decline in light oil reserves, the development of high-viscosity oil fields has become more relevant today. One of the most common methods of thermal enhanced oil recovery is cyclic-steam stimulation. Compared to other techniques, this method has an advantage of providing higher thermal efficiency due to selection of its parameters with account of variations in the basic properties of a heat carrier as it moves towards the well bottom. The purpose of this work is to create a method for rapid assessment of optimal technological parameters of cyclic-steam stimulation, in which the temperature distributions along boreholes obtained from geophysical research are used as initial data. The created method allows determination of the time of hot carrier injection, steam soak and oil production. The scientific novelty of the study refers to the fact that an integrated approach relating the problem of hot carrier transportation with the problem of determination of optimal parameters for reservoir treatment is proposed for the first time. The cyclic-steam stimulation process is described by the equations of mechanics of multiphase systems. Euler’s method is used to solve the resulting system of equations. The paper presents: functions of depth-wise distributions of pressure, temperature and velocity and dryness fraction of steam; temperature distribution in the multilayer wall of the well and in the rock (considering a specified heat transfer between the layers of the well materials, water and rock at known temperatures); optimal values of the time of hot carrier injection, steam soak and oil production. The calculations have shown that, due to the in the layer of basalt fiber, the temperature in the multilayer wall of the well decreases within 80%. The maximum value of additional cumulative oil recovery has been established by analyzing the calculated bottom-hole parameters of the heat carrier.

On the detection of nonlinear normal mode-related isolated branches of periodic solutions for high-dimensional nonlinear mechanical systems with frictionless contact interfaces

This contribution introduces a methodology for the detection of isolated branches of periodic solutions to the nonlinear mechanical equation of motion for systems featuring frictionless contact interfaces. This methodology relies on a harmonic balance method-based solving procedure combined with the application of the Melnikov energy principle. It is able to predict the location of isolated branches of solutions in the vicinity of families of autonomous periodic solutions, namely nonlinear normal modes. The methodology is first applied on a two-degree-of-freedom phenomenological system in order to illustrate its relevance and accuracy. In particular, for this academic application, the proposed methodology yields an understanding of the discontinuous evolution of the first nonlinear resonance frequency as the forcing amplitude increases. Isolated branches of solutions featuring high amplitudes of vibration are detected far beyond nonlinear resonance frequencies obtained with a typical application of the harmonic balance method coupled with an arc-length continuation algorithm. Demonstration of the applicability of the proposed methodology to high-dimensional industrial finite element models is made with the nonlinear vibration analysis of a transsonic compressor blade, NASA rotor 37, subjected to an harmonic loading and blade-tip/casing structural contacts. The proposed methodology yields numerous isolated branches of solutions, which relevance is assessed by means of time integration simulations. In the end, the presented results underline that typical continuation algorithms may yield a significant underestimation of nonlinear resonance frequencies.

An efficient computational framework for height-contained growing and intersecting hydraulic fracturing simulation via SGBEM–FEM

An efficient computational framework to model the growth of intersecting height-contained hydraulic fractures is proposed. The fracture in solids is modeled by the classical theory of linear elastic fracture mechanics, whereas the injection flow in hydraulic fractures is treated as channel flow. The governing equations of fracture mechanics are formulated in terms of weakly-singular weak-form boundary integral equations, with the knowledge of the fracture surface geometry exploited to reduce the integral dimension. The fluid flow within the fracture is treated using finite element method as one dimensional flow in an arbitrarily curved channel. The symmetric Galerkin boundary element method and finite element method are coupled with the traction continuity condition. The model is then formulated in a variational form and augmented with Lagrange multipliers to prescribe displacement constraints on intersections of hydraulic fractures, while fluid constraints are enforced by a consolidation of associated pressure degrees of freedom. Pertinent numerical implementation techniques, including spatial and temporal discretizations, special crack tip element, crack propagation criteria and remeshing strategy, are presented. The proposed framework is verified by comparing to the numerical results from a three dimensional hydraulic fracturing simulator and then verified by comparing the stress intensity factors to the XFEM and Murakami’s results for both star shaped cracks and symmetric branched cracks. The verification shows the utility properly enforces the physical constraints on the intersections of fractures. Three numerical examples are shown to demonstrate the capability of the proposed computational framework modeling the intersecting hydraulic fracturing process in a single or separated height-contained pay zones.

Sensorless control of dual three-phase permanent magnet synchronous motor based on speed feedback and frequency-variable tracking.

The dual three-phase Permanent Magnet Synchronous Motor (PMSM) control system is characterized by its high reliability, slight torque fluctuation and low harmonic content. It is very suitable for systems requiring high power output and high reliability, for instance, electric vehicles, aerospace and military equipment. In this paper, a full speed domain sensorless control technology for dual three-phase PMSM is proposed which solves the limitations of other sensorless controls, improves system accuracy and stability, and has high practicality in fields such as new energy vehicles. The mathematical model of this motor in a static coordinate system is established, and the sine and cosine signals along with the velocity and angle information are obtained by using the flux linkage observer. Moreover, the estimated angle error parameter is introduced into the flux linkage observer; as a result, the estimation accuracy is improved by the estimated speed feedback, and the current frequency is tracked by the stator current Frequency-Variable Tracker (FVT) to reduce the current error. Meanwhile, to make the observer's estimation more accurate and to improve its ability to resist disturbance, a rotor disturbance is added to act as a disturbance variable. Through the mechanical motion equation of the motor, a fourth-order Extended State Observer (ESO) is built to calculate the rotor position and speed. Finally, the technology accuracy is verified using simulation and experimental results. The findings prove that the sensorless detection technology, with speed feedback introduce in this paper, has good reliability and high precision for dual three-phase PMSM under dynamic and static conditions.

Multi-temporal decomposition for elastoplastic ratcheting solids

This paper presents a multi-temporal formulation for simulating elastoplastic solids under cyclic loading. We leverage the proper generalized decomposition (PGD) to decompose the displacements into multiple time scales, separating the spatial and intra-cyclic dependence from the inter-cyclic variation. In contrast with the standard incremental approach, which solves the (non-linear and computationally intensive) mechanical balance equations at every time step, the proposed PGD approach allows the mechanical balance equations to be solved exclusively for the small-time intra-cyclic response, while the large-time inter-cyclic response is described by simple scalar algebraic equations. Numerical simulations exhibiting complex cyclic responses, including a 2D problem and an application to a monopile foundation, demonstrate that PGD solutions with a limited number of space–time degrees of freedom may be obtained numerically, only requiring a few modes to accurately capture the reference response.