Stationary Equation Research Articles

A selfdual variational principle with minimal hypothesis and applications to stationary, dynamic and stochastic equations

We revisit the self-dual variational approach to the resolution of non-linear stationary, parabolic and stochastic partial differential equations involving a monotone non-linearity, by relaxing the hypothesis under which it is applicable, in two directions: The prohibitive coercivity condition which was problematic for applications to evolution equations, be them deterministic or stochastic, and the restrictive “regularity” property that required the domain of the non-linear operator under study to be large. The new principle is applied to give more direct proofs for the existence of weak solutions for incompressible Navier-Stokes equations in their stationary, dynamic or stochastic versions.

Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition

We study the asymptotic behavior of solutions to the steady Navier-Stokes equations outside of an infinite cylinder in R3. We assume that the flow is periodic in x3-direction and has no swirl. This problem is closely related with two-dimensional exterior problem. Under a condition on the generalized finite Dirichlet integral, we give a pointwise decay estimate of the vorticity at the spatial infinity. This reveals the relation between the integrability of ∇v and the decay rate of ω near the spatial infinity. Moreover, we prove a Liouville-type theorem only from the condition of the generalized finite Dirichlet integral.

An iterative method for the thermally coupled incompressible magnetohydrodynamics equations at high parameter

We design an iterative method based on linearization approach for the thermally coupled stationary incompressible magnetohydrodynamics equations at high physical parameters. The iterative method includes solving the considered equations by the linearization approach and applying the error correction strategy to reduce the iterative error arising from the linearization approach for solving the nonlinear equations. The iterative method not only can improve computational capability for solving the considered equations at high physical parameters but also can save computational time and iterative step. Moreover, we prove the stability of the iterative method and derive the iterative error estimates. Finally, some numerical tests are implemented to illustrate the efficiency of the presented method.

New static solutions of symmetric [formula omitted] equation

In this paper, we provide new exact solutions of nonlinear Klein–Gordon (ϕ4) equation in 1+1-dimension. For simplicity, we focus on the static equation and ignore the time-dependence. The symmetric ϕ4 equation has played an important role in many areas of physics. We obtain several novel non-singular solutions of the symmetric ϕ4 model in terms of the Jacobi elliptic functions and compare them with the well-known solutions. Finally, we categorize these solutions in terms of the potential parameters.

Non-isothermal non-Newtonian fluids: The stationary case

The stationary Navier–Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet-type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suitable power law depending on [Formula: see text] (shear thinning case). For this problem we establish the existence of a weak solution as well as we prove some regularity results both for the Navier–Stokes and the Stokes cases. Then, the latter case with the Carreau power law is approximated through a FEM scheme and some error estimates are obtained. Such estimates are then validated through some two-dimensional numerical experiments.

A Liouville-type theorem for the stationary MHD equations

We establish a new Liouville-type theorem for solutions of the stationary MHD equations imposing asymmetric oscillation growth conditions on the tensor-valued functions for the velocity and the magnetic field.

Monitoring method of bolt looseness based on inversion of joint surface gap by magnetic field change

In the field of engineering, bolted connection structures are widely used, and loose bolts can lead to serious safety incidents. However, current monitoring techniques for bolt loosening are limited in their ability to detect the degree of looseness in real-time. To address this problem, this study proposes a novel approach to monitoring bolt loosening by using the inversion of magnetic field change combined with surface clearance. Firstly, the static magnetic field equation is theoretically derived using the Maxwell equation under the assumption of no current, and the sensitivity of the magnetic field to the gap is examined using finite element simulation. Secondly, a three-dimensional finite element model is established to conduct a parametric scan of the gap, and the relationship between the gap and magnetic field intensity is obtained. Furthermore, the study proposes a high-precision gap sensor design scheme, which is theoretically confirmed to be viable. To confirm the feasibility of the proposed bolt looseness monitoring method and the instrument development, a bolt looseness testing platform is built and applied in engineering practice. The research shows that this technique can successfully track the state of loosening bolts, offering a new approach and concept for bolt loosening monitoring. It has the potential for a wide range of applications.

Static and dynamic analysis of the postbuckling of axially moving spin beams

This paper aims to investigate the static and dynamic behaviors of the postbuckling of axially moving spin beams (bi-gyro system). The Hamilton’s principle is used to derive the nonlinear governing equations of bi-gyro systems, which are based on the Euler-Bernoulli beam model and account for von Karman geometric nonlinearity. After neglecting the time-dependent terms, the static equations are discretized using the differential quadrature method and then solved as a nonlinear algebraic system via the Newton-Raphson iteration method. Subsequently, the linear vibration problem is accurately solved for axially moving spinning beams in their first buckled configuration. Finally, numerical calculations are conducted to investigate the impact of axial and spinning velocities on the steady-state dynamic responses of bi-gyro systems. The findings are as follows: (a) the critical axial velocity where trivial equilibrium is unstable remains unaffected by the presence of spinning velocity; (b) the bifurcation point of the axial velocity corresponding to amplitude is reduced by spinning velocity, resulting from centrifugal force; (c) the vibration amplitude of the bi-gyro system will diverge under a weaker displacement constraint spring; and (d) stable buckled and straight configurations can coexist under certain axial and spinning velocities.

Two‐level stabilized finite volume method for the stationary incompressible magnetohydrodynamic equations

AbstractIn this paper, a two‐level stabilized finite volume method is developed and analyzed for the steady incompressible magnetohydrodynamic (MHD) equations. The linear polynomial space is used to approximate the velocity, pressure and magnetic fields, and two local Gauss integrations are introduced to overcome the restriction of discrete inf‐sup condition. Firstly, the existence and uniqueness of the solution of the discrete problem in the stabilized finite volume method are proved by using the Brouwer's fixed point theorem. ‐stability results of numerical solutions are also presented. Secondly, optimal error estimates of numerical solutions in and ‐norms are established by using the energy method and constructing the corresponding dual problem. Thirdly, the stability and convergence of two‐level stabilized finite volume method for the stationary incompressible MHD equations are provided. Theoretical findings show that the two‐level method has the same accuracy as the one‐level method with the mesh sizes . Finally, some numerical results are provided to identify with the established theoretical findings and show the performances of the considered numerical schemes.

The installation load prediction method of interference fit between the submarine elliptical pipeline and mechanical connector

For damaged submarine pipelines, interference fit is necessary for the repair technology by the mechanical connector. Due to manufacturing error and asymmetric load, the ovality occurs on pipeline and increases the difficulty to install the connector. For ensure safety, it is necessary to consider the interference and ovality of pipeline to predict the installation load. In this paper, the software ANSYS was used to establish the finite element model of the connector and pipeline, then the contact characteristics on the contact surface were analyzed. According to the static equilibrium equation, the installation load of interference fit is equivalent to the friction force and is proportional to the contact pressure, it could be predicted according to the contact pressure. The results indicate that, on the premise of the same equivalent interference, the contact force and contact area on the contact surface were independent of the ovality of pipeline. For the pipelines with different ovality, the installation load can be predicted based on contact force of the circular pipeline. Some cases are conducted to verify the accuracy of the established prediction formula. The load prediction formula effectively solves the problem of submarine pipeline repair connection and has high engineering application value.

Electrochemical performance and impedance of a conical pore in the low–Pt PEM fuel cell catalyst layer

AbstractA model for the transient electrochemical performance of a conical pore in the cathode catalyst layer of a low–Pt PEM fuel cell is developed. The pore is separated from the Pt surface by a thin ionomer film. A transient equation for the oxygen diffusion along the pore coupled to the proton conservation equation in the ionomer film is derived. Numerical solution of the static equations shows superior electrochemical performance of a conical pore as compared to cylindrical pore with equivalent electrochemically active surface area. Equations for the pore impedance are derived by linearization and Fourier–transform of transient equations. The conical pore impedance is calculated and compared to the impedance of equivalent cylindrical pore. It is shown that the pore shape affects the frequency dependence of impedance.

Machine Learning Algorithms for Identifying Dependencies in OT Protocols

This study illustrates the utility and effectiveness of machine learning algorithms in identifying dependencies in data transmitted in industrial networks. The analysis was performed for two different algorithms. The study was carried out for the XGBoost (Extreme Gradient Boosting) algorithm based on a set of decision tree model classifiers, and the second algorithm tested was the EBM (Explainable Boosting Machines), which belongs to the class of Generalized Additive Models (GAM). Tests were conducted for several test scenarios. Simulated data from static equations were used, as were data from a simulator described by dynamic differential equations, and the final one used data from an actual physical laboratory bench connected via Modbus TCP/IP. Experimental results of both techniques are presented, thus demonstrating the effectiveness of the algorithms. The results show the strength of the algorithms studied, especially against static data. For dynamic data, the results are worse, but still at a level that allows using the researched methods to identify dependencies. The algorithms presented in this paper were used as a passive protection layer of a commercial IDS (Intrusion Detection System).

Meshless generalized finite difference method for two- and three-dimensional transient elastodynamic analysis

In this paper, a meshless collocation method is introduced for two-dimensional (2D) and three-dimensional (3D) transient elastodynamic problems by applying the generalized finite difference method (GFDM) in conjunction with the Houbolt scheme. Coupled equilibrium equations with a time-dependent loading are transformed into the static equations at time nodes by adopting the Houbolt method. After then, the solution of the static equations is achieved with the GFDM with second-order and fourth-order expansions. Several numerical examples involving complicated geometries and different initial and boundary conditions are simulated to validate the performance of the present approach.

Optimization of the Clamping Layout of Thin-walled Parts with Partition Frame

When the thin-walled parts are machined, they need to be clamped and positioned. By using finite element method (FEM), the deformation law of workpiece clamping caused by single clamping layout parameter can be gained. However, it is impossible to express the relationship between the clamping deformation and the clamping layout parameter by FEM when multiple clamping layout parameters are taken into account simultaneously. Therefore, ABAQUS is used to establish a three-dimensional finite element model for the clamping layout of thin-walled part. And by combining the milling force and the static balance equation, the appropriate clamping force value can be obtained. Then it is convenient to obtain the initial data sample of genetic algorithm. Genetic algorithm is used to optimize the clamping layout of frame type of thin-walled part and obtains the optimal layout. The minimum value of maximum clamping deformation is 15.50 µm. In the field of thin-walled part machining, the research results of this paper have an important guiding role in the rational design of the clamping layout scheme.

The Problem of Complex Heat Transfer with Cauchy-Type Conditions on a Part of the Boundary

The paper considers a boundary value problem for stationary equations of complex heat transfer with an undetermined boundary condition for the radiation intensity on a part of the boundary and an overdetermined condition on another part of the boundary. An optimization method for solving this problem is proposed, and an analysis of the corresponding problem of boundary optimal control is presented. It is shown that the sequence of solutions of extremum problems converges to the solution of a problem with Cauchy-type conditions. The efficiency of the algorithm is illustrated by numerical examples.

Development of a mathematical model for a compound technological complex of vanyukov melting in order to control the material and thermal regime

This article presents a mathematical model in the form of static equations of dependencies of input and output flows based on the equations of material and heat balance for the purposes of operational planning and control of the complex technological complex of Vanyukov melting (PV). Dynamic characteristics are presented for the purpose of controlling the thermal regime based on the technology of the developed melting process with blowing from below. As a result of the study, the developed mathematical model for controlling the smelting process when calculating the material flows of the charge will allow tracking changes in the thermal state of the smelting (by the copper content in the matte). This model can quite well describe the dynamics of the state of the process, both when establishing the impacts aimed at increasing the heating of the furnace, and at reducing its heating. Based on the equations, a computer model based on the dynamic programming method in the MATLAB software package has been developed. The scientific novelty lies in the fact that for the first time, the structure of a mathematical model has been developed that describes the processes occurring in the over-tuyere zone and the sludge zone of the smelting products.

Proof of the Concept of Detailed Dynamic Thermal-Hydraulic Network Model of Liquid Immersed Power Transformers

The paper presents a physics-based method to calculate in real time the distribution of temperature in the active part of liquid immersed power transformers (LIPT) in a transient thermal processes during grid operation. The method is based on the detailed dynamic thermal-hydraulic network model (THNM). Commonly, up to now, lumped models have been used, whereby the temperatures are calculated at a few points (top-oil and hot-spot), and the parameters are determined from basic or extended temperature-rise tests and/or field operation. Numerous simplifications are made in such models and the accuracy of calculation decreases when the transformer operates outside the range of tested values (cooling stage, loading). The dynamic THNM reaches the optimum of accuracy and simplicity, being feasible for on-line application. The paper presents fundamental equations of dynamic THNM, which are structurally different from static THNM equations. The paper offers the numerical solver for the case of a closed-loop thermosyphon. To apply the method for real transformer grid operation, there is a need to develop details as in static THNM, which has been used to calculate the distribution of the temperatures in LIPT thermal design. The paper proves the concept of dynamic THNM using the experimental results of a closed-loop thermosyphon small-scale model, previously published by authors from McGill University in 2017. The comparison of dynamic THNM with measurements on that model are presented in the paper.

Regularity of Hamiltonian stationary equations in symplectic manifolds

In this paper, we prove that any C1-regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. More broadly, we develop a regularity theory for a class of fourth order nonlinear elliptic equations with two distributional derivatives. Our fourth order regularity theory originates in the geometrically motivated variational problem for the volume functional, but should have applications beyond.

Physics-informed machine learning and stray field computation with application to micromagnetic energy minimization

We study the full 3d static micromagnetic equations via a physics-informed neural network (PINN) ansatz for the continuous magnetization configuration. PINNs are inherently mesh-free and unsupervised learning models. In our approach we can learn to minimize the total Gibbs free energy with additional conditional parameters, such as the exchange length, by a single low-parametric neural network model. In contrast, traditional numerical methods would require the computation and storage of a large number of solutions to interpolate the continuous spectrum of quasi-optimal magnetization configurations. We also consider the important and computationally expensive stray field problem via PINNs, where we use a basically linear learning ansatz, called extreme learning machines (ELM) within a splitting method for the scalar potential. This reduces the stray field training to a linear least squares problem with precomputable solution operator. We validate the stray field method by means of numerical example comparisons from literature and illustrate the full micromagnetic approach via the NIST μMAG standard problem #3.

Nonlocal reduced integrable mKdV-type equations from a vector integrable hierarchy

This paper aims to present two hierarchies of nonlocal reduced integrable mKdV-type equations from a vector integrable hierarchy associated with a matrix Lie algebra, not being A type. The key point is to make similarity transformations for the spectral matrix, which keep the associated zero curvature equations invariant and then there follow reduced nonlocal integrable mKdV-type equations. The success lies in determining a Laurent series solution to the corresponding reduced stationary zero curvature equation, which generates temporal matrix spectral problems in the zero curvature formation.