Stationary Equation Research Articles

Numerical modeling of discharge characteristics of a planar magnetron with injection of electrons from auxiliary discharge

We present here a numerical model of a planar magnetron with electron injection from the plasma of an auxiliary discharge with the hollow cathode. The model is based on solving a system of stationary equations using the method of successive approximations. The model takes into account the effects of the current of the injected electrons, the operating gas pressure, the magnetic field induction, the flow of the target material into the plasma, and the change in the plasma mass-to-charge composition due to cathode sputtering by ion bombardment. Typical volt-ampere characteristics of a magnetron discharge with electron injection, as well as the mass-to-charge composition of the created gas-metal plasma, are calculated.

RPINNs: Rectified-physics informed neural networks for solving stationary partial differential equations

Due to the development of high performance computing, deep learning algorithm has made a significant progress in many fields such as computational mathematics. The physics-informed neural networks have put forward a innovative idea for tackling a broad range of forward and inverse problems of partial differential equations. Motivated by the philosophy of physics-informed neural network and the multigrid method, we introduce the gradient information of numerical solution of physics-informed neural network into the new neural networks and propose the rectified-physics informed neural network for solving stationary partial differential equations. And for solving multi-objective optimization of neural networks, the dynamic weight strategy is adopted to balance numerical difference among terms in the loss function, and effectively alleviate the gradient ill-conditioned phenomenon. Finally, we perform a series of numerical experiments to demonstrate effectiveness of the RPINNs method which is combined with the dynamic weight strategy to improve calculation accuracy.

Digital Analysis of Common Turbulence Patterns in Centrifugal Pump Flow Simulation Based on COMSOL Multiphysics Software

The present paper tackles the convergence and performance of three numerical turbulence models in the flow simulation. The benchmark analysis was performed using the COMSOL Multiphysics code, and the turbulence on a centrifugal water pump was generated numerically using the k−ε,k−ω, and k−ω SST models. However, the geometry was conducted on SolidWorks due to its complexity. First, the flow modeling was driven by solving the stationary Navier-Stokes equations. Then, the effects of the tested models, on the numerical CFD simulation, were examined. The analyzed results demonstrated that the best calculation precision was obtained using the k−ω model, whereas the lowest was provided by the k−ω SST model. A remarkable pumping performance was also recorded.

On calculation of stress-strain state of steel closed ropes in extension and twisting. Part 2. Influence of torsion on the stress-strain state of closed rope

Comparative analysis of the results of determination of stress-strain state in a closed lifting rope during extension and twisting was conducted via analytical calculation according to the linear static equations and computer-aided finite element modeling. Computer modeling allows to establish the features of variation of internal axial forces and torques in the layers that are laid with different directions. It makes possible to predict workability of a closed lifting rope during operation. The most intensive redistribution of internal forces and torques in the internal and subsurface layers during rope turns under the effect of external torque was found out in comparison with extension. Decrease of the torque and axial force values to zero in the deposited external layer during its untwisting testifies about switching this layer off the operation and possibility of rope lamination in the case of external pressure absence. Internal compression of subsurface layer from the side of external rope layer, in its turn, decreases possibility of stability loss (opening) of this layer during its twisting. Deviation from the common rule for variation features of relative elongation ε during joint extension and twisting was shown during modeling. So, external untwisting torque tries to extend the rope (increase of ε), while twisting torque tries to compress it (decrease of ε), what is connected with influence of gap values between the rope elements (wires and layers) on distribution of the contact stresses between them during twisting.

Deterministic and Discriminative Imitation (D2-Imitation): Revisiting Adversarial Imitation for Sample Efficiency

Sample efficiency is crucial for imitation learning methods to be applicable in real-world applications. Many studies improve sample efficiency by extending adversarial imitation to be off-policy regardless of the fact that these off-policy extensions could either change the original objective or involve complicated optimization. We revisit the foundation of adversarial imitation and propose an off-policy sample efficient approach that requires no adversarial training or min-max optimization. Our formulation capitalizes on two key insights: (1) the similarity between the Bellman equation and the stationary state-action distribution equation allows us to derive a novel temporal difference (TD) learning approach; and (2) the use of a deterministic policy simplifies the TD learning. Combined, these insights yield a practical algorithm, Deterministic and Discriminative Imitation (D2-Imitation), which oper- ates by first partitioning samples into two replay buffers and then learning a deterministic policy via off-policy reinforcement learning. Our empirical results show that D2-Imitation is effective in achieving good sample efficiency, outperforming several off-policy extension approaches of adversarial imitation on many control tasks.

Stability of stationary subsonic compressible Euler flows with mass-additions in two-dimensional straight ducts

We show existence, uniqueness and stability for a family of stationary subsonic compressible Euler flows with mass-additions in two-dimensional rectilinear ducts, subjected to suitable time-independent multi-dimensional boundary conditions at the entrances and exits. The stationary subsonic Euler equations consist a quasi-linear system of elliptic-hyperbolic composite-mixed type, while addition-of-mass destructs the usual methods based upon conservation of mass and Lagrangian coordinates to separate the elliptical and hyperbolic modes of the system. We establish a new decomposition and nonlinear iteration scheme to overcome this major difficulty. It reveals that mass-additions introduce very strong interactions in the elliptic and hyperbolic modes, and lead to a class of second-order elliptic equations with multiple integral nonlocal terms. The linearized problem is solved by studying algebraic and analytical properties of infinite weakly coupled boundary-value problems of ordinary differential equations, each with multiple nonlocal terms, after applications of Fourier analysis methods.

Asymptotic series solution of variational stokes problems in planar domain with crack-like singularity

ABSTRACT A class of nonlinear variational problems describing incompressible fluids and solids by stationary Stokes equations given in a planar domain with a crack (infinitely thin flat plate in fluids) is considered. Based on the Fourier asymptotic analysis, general analytical solutions are obtained in polar coordinates as the power series with respect to the distance to the crack tip. The logarithm terms and angular functions are accounted in the asymptotic expansion using recurrence relations. Then boundary conditions imposed between the opposite crack faces in the sector of angle determine admissible exponents and parameters in the power series. For the specific conditions of Dirichlet, Neumann, impermeability, non-penetration and shear crack, the principal asymptotic terms are derived, which verify the singular behaviour. In particular, the analytical solution answers the questions of a square-root singularity at the crack tip and the presence of log-oscillations of variational solutions for the Stokes problems.

Singular extinction profiles of solutions to some fast diffusion equations

We study extinction profiles of solutions to fast diffusion equations with some initial data in the Marcinkiewicz space. The extinction profiles will be the singular solutions of their stationary equations.

Partially regular weak solutions of the stationary Navier-Stokes equations in dimension 6

By using defect measures, we prove the existence of partially regular weak solutions to the stationary Navier-Stokes equations with external force f in L_{text {loc}}^q cap L^{3/2}, q>3 in general open subdomains of {mathbb {R}}^6. These weak solutions satisfy certain local energy estimates and we estimate the size of their singular sets in terms of Hausdorff measures. We also prove the defect measures vanish under a smallness condition, in contrast to the nonstationary Navier-Stokes equations in {mathbb {R}}^4 times [0,infty [.

Thermochemical Analysis of a Packed-Bed Reactor Using Finite Elements with FlexPDE and COMSOL Multiphysics

This work presents the thermochemical analysis of a packed-bed reactor via multi-dimensional CFD modeling using FlexPDE and COMSOL Multiphysics. The temperature, concentration, and reaction rate profiles for methane production following the Fischer–Tropsch (F-T) synthesis were studied. To this end, stationary and dynamic differential equations for mass and heat transfer were solved via the finite element technique. The transport equations for 1-D and 2-D models using FlexPDE consider dispersion models, where the fluid and the catalyst can be treated as either homogeneous or heterogenous systems depending on the gradient extents. On the other hand, the 3-D model obtained in COMSOL deems the transport equations incorporated in the Porous Media module. The aim was to compare the FlexPDE and COMSOL models, and to validate them with experimental data from literature. As a result, all models were in good agreement with experimental data, especially for the 2-D and 3-D dynamic models. In terms of kinetics, the predicted reaction rate profiles from the COMSOL model and the 2-D dynamic model followed the temperature trend, thus reflecting the temperature dependence of the reaction. Based on these findings, it was demonstrated that applying different approaches for the CFD modeling of F-T processes conducts reliable results.

O. A. Ladyzhenskaya’s System of Equations of Symmetric Boundary Layer of Modified Fluid

In this paper, we study the system of equations of a boundary layer for a nonlinearly viscous, electrically conductive liquid described by a rheological law proposed by O. A. Ladyzhenskaya for incompressible media. The boundary-layer equations for the Ladyzhenskaya model were first obtained from Prandtl’s axioms. By the Mises transform, the system of boundary-layer equations can be reduced to a single quasilinear equation. The main method used in this paper is the Crocco transform, which turns the system of boundary-layer equations into a quasilinear degenerate parabolic equation. In contrast to the Mises variables, the Crocco substitution allows one to study both stationary and nonstationary equations.

Ground deformation as a result of the foundation of a new module in the Spanish Antarctic Base Gabriel de Castilla, Deception Island, Antarctica

Since its inauguration in the 1989/90 campaign, the Spanish Antarctic Base Gabriel de Castilla (BAEGC) has supported an increasing volume of scientific projects, stimulating the expansion of its facilities. The permafrost soil on which they are located is affected by changing thermal and periglacial conditions that disturb its mechanical and physical characteristics. This paper analyzes the foundation behavior of a building (18.5x6.5x2.7 m length – width – height, around 104 kg in operation supported on 16 posts) through the geomechanical study (stresses and deformations) of the ground during the winter and southern summer seasons, supported by three types of foundations: piles, footings and slabs. These are commonly recommended as support elements inperiglacial soils affected by thermal-seasonal variations. The geotechnical properties have been obtained from stress-tests on soil samples collected in the surroundings of the BAEGC, General Marvá Army Engineers Laboratory (INTA). Considering the thermal variations in the tests on frozen and unfrozen samples, the geotechnical column type has been established and extended along the profile of the 2D domain. The stationary stress-strain equations with a constitutive Mohr-Coulomb rupture model have been solved using finite elements in MIDAS GTS-NX, for each season and each foundation, discretizing with a triangular or quadrangular mesh adapted to the 2D domain geometry with the foundation. The numerical results show that, in any tested case, there is no critical ground failure, and the foundation with the smallest displacement (about 0.036 m), both in summer and winter, is the 1.2 m foundation placed at 0.5 m depth on permafrost.

Penalty-Factor-Free Stabilized Nonconforming Finite Elements for Solving Stationary Navier-Stokes Equations

Penalty-Factor-Free Stabilized Nonconforming Finite Elements for Solving Stationary Navier-Stokes Equations

A Long-Stroke Nanopositioning Stage With Annular Flexure Guides

This article presents a long-stroke nanopositioning stage supported with annular flexure guides to be used in high-precision optical instruments. First, the structure of the novel annular flexure guide and the positioning stage is proposed. Because of the symmetric configuration of the flexure guide, large motion range, high linearity, and high stiffness ratio of the positioning stage are achieved. Second, the stiffness of the flexure guide is modeled by applying the two-port mechanical network method combining with the Castigliano’s theorem and the static equilibrium equation of the structure. Afterwards, the dimension parameters of the flexure guide are optimized by applying the stiffness model to maximize the motion stiffness and the stiffness ratio of the positioning stage. Model validation and performance evaluation are conducted via finite element analysis. Finally, a prototype of the nanopositioning stage is fabricated, and the classical feedforward proportional integral derivative (PID) controller is employed to improve its motion accuracy. Experimental results show that the proposed positioning stage can achieve <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pm$</tex-math></inline-formula> 5 mm motion range, 20 nm motion resolution, 20 nm positioning accuracy at the maximum output position, and micron tracking accuracy for both sinusoidal trajectory and triangular trajectory under the closed-loop controller.

Vibration and Constitutive Relations of Viscoelastic Cantilever Beam of HTPB Propellant

A cantilever beam test specimen of a viscoelastic material, which is sourced from the hydroxy-terminated polybutadiene (HTPB) propellant, has been made. The traditional experimental modal analysis (EMA) is performed using the SIEMENS TESTLAB software to obtain the first six orders of modal frequencies based on the Poly MAX method, thereby determining the frequency range associated with the frequency domain analysis. The relaxation modulus of this viscoelastic material is first measured using a thermo-viscoelasticity analyzer (DMA); subsequently, the relaxation modulus is adopted to fit the 19-parameter Prony series of conformational models that characterize this viscoelastic material. Sinusoidal frequency sweep is then conducted on the SIEMENS TESTLAB platform, and the corresponding measured signals are obtained. Using the finite element method, a transient analysis simulation of the experimental conditions is performed. The dynamic response of this viscoelastic material under sinusoidal frequency sweep is evaluated by comparing it with the experimental data. It is observed that the overall acceleration peak error and the finite element simulation result are less than 3%. This study verifies that the obtained static constitutive equation (19-parameter Prony series) can describe the dynamic responses at low frequencies (&lt;80 Hz).

Uniaxial polarization analysis of bulk ferromagnets: theory and first experimental results.

On the basis of Brown's static equations of micromagnetics, the uniaxial polarization of the scattered neutron beam of a bulk magnetic material is computed. The approach considers a Hamiltonian that takes into account the isotropic exchange interaction, the antisymmetric Dzyaloshinskii-Moriya interaction, magnetic anisotropy, the dipole-dipole interaction and the effect of an applied magnetic field. In the high-field limit, the solutions for the magnetization Fourier components are used to obtain closed-form results for the spin-polarized small-angle neutron scattering (SANS) cross sections and the ensuing polarization. The theoretical expressions are compared with experimental data on a soft magnetic nanocrystalline alloy. The micromagnetic SANS theory provides a general framework for polarized real-space neutron methods, and it may open up a new avenue for magnetic neutron data analysis on magnetic microstructures.

Contingency analysis for gas transport networks with hydrogen injection

Novel methods for contingency analysis of gas transport networks are presented. They are motivated by the transition of our energy system where hydrogen plays a growing role. The novel methods are based on a specific method for topological reduction and so-called supernodes. Stationary Euler equations with advanced compressor thermodynamics and a gas law allowing for gas compositions with up to 100% hydrogen are used. Several measures and plots support an intuitive comparison and analysis of the results. In particular, it is shown that the newly developed methods can estimate locations and magnitudes of additional capacities (injection, buffering, storage etc.) with a reasonable performance for networks of relevant composition and size.

Forward Kinematics for Suspended Under-Actuated Cable-Driven Parallel Robots With Elastic Cables: A Neural Network Approach

Abstract Kinematic analysis of under-constrained cable-driven parallel robots (CDPR) has been a topic of interest because of the inherent coupling between the loop-closure and static equilibrium equations. The non-linearity of the problem is magnified with the addition of the coupling between the cable lengths and their tensions based on the elastic cable model. The paper proposes an unsupervised neural network algorithm to perform real-time forward geometrico-static analysis of such robots in a suspended configuration under the action of gravity. The formulation determines a non-linear function approximation to model the problem and proves to be efficient in solving consecutive and close waypoints in a path. The methodology is applied on a six-degree-of-freedom spatial under-constrained suspended CDPR. Specific comparison results in simulation and hardware to show the effectiveness of the proposed method in tracking a given path are illustrated. Finally, the degree of constraint satisfaction is presented against the results obtained from non-linear least-square optimization.

Application of systematic error detection method to integral parameter measurements in sophisticated production processes and operations

This paper uses the Lagrange multiplier method to substantiate the general solution for validity control problem in terms of independently measured parameters of material flows. The paper shows that the efficiency of the above method can be raised if certain process parameters are measured in a number of independent ways by a dispatcher control system. A technique is described to adjust the initial measurement signals in case some channels have redundant information. Such technique is applicable if the values to be adjusted are identified over a period of time that greatly exceeds the dynamic memory of the object or if the dynamic links between certain channels are completely compensated and the execution of static equations of the studied processes is a prerequisite for the values to be adjusted. A method for systematic error detection in application to aggregate integral uncorrelated measurements was used as part of the NavoiAzot dispatcher control system for adjustment of per-shift and per-day performance indicators as calculated by the system. The paper considers the problem of selecting and retrieving valid data on the basis of redundant measurements of material flows and flow constraint equations. The paper describes a solution for the problem of selecting and retrieving valid data on the basis of redundant measurements of material flows and flow constraint equations in application to a complex fertilizer production monitoring system.

Solution to the two-body Smoluchowski equation with shear flow for charge-stabilized colloids at low to moderate Péclet numbers

We developed an analytical theoretical method to determine the microscopical structure of weakly to moderately sheared colloidal suspensions in dilute conditions. The microstructure is described by the static structure factor, obtained by solving the stationary two-body Smoluchowski advection-diffusion equation. The singularly perturbed PDE problem is solved by performing an angular averaging over the extensional and compressing sectors and by the rigorous application of boundary-layer theory (intermediate asymptotics). This allows us to expand the solution to a higher order in P\'eclet with respect to previous methods. The scheme is independent of the type of interaction potential. We apply it to the example of charge-stabilized colloidal particles interacting via the repulsive Yukawa potential and study the distortion of the structure factor. It is predicted that the distortion is larger at small wavevectors $k$ and its dependence on $Pe$ is a simple power law. At increasing $Pe$, the main peak of the structure factor displays a broadening and shift towards lower $k$ in the extensional sectors, which indicates shear-induced spreading out of particle correlations and neighbor particles locally being dragged away from the reference one. In the compressing sectors, instead, a narrowing and shift towards high $k$ is predicted, reflecting shear-induced ordering near contact and concomitant depletion in the medium-range. An overall narrowing of the peak is also predicted for the structure factor averaged over the whole solid angle. Calculations are also performed for hard spheres, showing good overall agreement with experimental data. It is also shown that the shear-induced structure factor distortion is orders of magnitude larger for the Yukawa repulsion than for the hard spheres.