Stationary Equation Research Articles

Two-level Newton iterative method based on nonconforming finite element discretization for 2D/3D stationary MHD equations

In this paper, two-level Newton iterative method based on nonconforming finite element discretization is presented for solving 2D/3D stationary incompressible magneto-hydrodynamics equations. First, the Crouzeix–Raviart type element for the velocity, and the conforming finite element for the magnetic field and pressure. The main idea of the proposed method is to solve MHD system by m Newton iterations on a coarse mesh, once correction by Stokes iteration on a fine mesh. The proposed method can save more computational time than one level method on the fine mesh with the same convergence rate. Moreover, the technical analysis of stability and optimal error estimates for two-level Newton iterative method are given. Finally, the applicability and efficiency of our proposed algorithm are illustrated by several numerical experiments.

Analysis of a hollow piezoelectric semiconductor composite cylinder under a thermal loading

Piezoelectric semiconductors (PSs) are the most promising building blocks for multifunctional electronic devices due to the polarization field resulting from piezoelectricity and pyroelectricity. In this article, we investigate the electromechanical coupling behaviors of a hollow PS composite (HPSC) cylinder under a uniform temperate change. The HPSC cylinder is consist of an outer thin PS layer and an inner elastic layer. Taking the interactions of thermoelasticity, piezoelectricity, and pyroelectricity into account, we develop one-dimensional equations for the HPSC cylinder with axisymmetric deformation, and present analytical solutions of the physical fields in the HPSC cylinder with perfect and imperfect interfaces for different boundary conditions. For the static equilibrium equation of the PS layer, the approximate relationship between the electric field and the average electric field is employed. The effects of geometrical parameters and imperfect interface, as well as the tuning effect of thermal loadings, on the multifield coupling responses and piezotronic effects of the HPSC cylinder are discussed in detail. The numerical results show that the physical fields are sensitive to the thermal loading, and the geometrical parameters, boundary conditions, and the interface bonding condition play an important role in the design of PS devices.

ЛИНЕЙНАЯ ТЕОРИЯ УПРУГОСТИ С УЧEТОМ КВАДРАТИЧНЫХ СЛАГАЕМЫХ

We suggest a linear theory version based on Taylor decompositions for stresses and power-series for quadratic summand deformations. Thus, static equations of equilibrium in stresses are written in the form of the second-order partial derivatives differential equations. The resolving equations of equilibrium in displacements are represented in the form of the third order partial derivatives differential equations. The physical equations in this version of the linear theory of elasticity are written in the same way as in the classical linear theory of elasticity. Equilibrium equations, along with other parameters – physical constants of the medium – contain minor parameters dx, dy, dz, the value of which, as shown by numerical modelling, has little effect on the nature of the stress-strain state. It is suggested to use experimental data to determine them. Along with the formulating of the basic equations of the three-dimensional theory of elasticity, particular cases of the stress-strain state of elastic continuous medium are considered: uniaxial stressed state; uniaxial deformed state; flat deformation; generalized plane stress state. Determination of the stressed and deformed state of a thin elastic bar by integrating the resolving equations in stresses and displacements is considered as examples. The suggested version of the linear theory of elasticity, due to the quadratic summand in Taylor decompositions for stresses and in power-series for deformations, expands the classical linear theory of elasticity and, with an appropriate experimental justification, can lead to new qualitative effects in the calculation of elastic deformable bodies.

An Improved Algorithm for the Retrieval of the Antarctic Sea Ice Freeboard and Thickness from ICESat-2 Altimeter Data

ICESat-2 altimeter data could be used to estimate sea ice freeboard and thickness values with a higher measuring accuracy than that achievable with data provided by previous altimeter satellites. This study developed an improved algorithm considering variable lead proportions based on the lowest point method to derive the sea surface height for the retrieval of Antarctic sea ice freeboard and thickness values from ICESat-2 ATL-07 data. We first collocated ICESat-2 tracks to corresponding Sentinel-1 SAR images and calculated lead (seawater) proportions along each track to estimate the sea surface height in the Antarctic Ocean. Then, the Antarctic sea ice freeboard and thickness were estimated based on a local sea surface height reference and a static equilibrium equation. Finally, we assessed the accuracy of our improved algorithm and ICESat-2 data product in the retrieval of the Antarctic sea ice thickness by comparing the calculated values to ship-based observational sea ice thickness values acquired during the 35th Chinese Antarctic Research Expedition (CHINARE-35). The results indicate that the Antarctic sea ice freeboard estimated with the improved lowest point method was slightly larger than that estimated with the ICESat-2 data product algorithm. The root mean squared error (RMSE) of the improved lowest point method was 35 cm with the CHINARE-35 measured sea ice thickness, which was smaller than that determined with the ICESat-2 data product algorithm (65 cm). Our improved algorithm could provide more accurate data on the Antarctic sea ice freeboard and thickness, thus supporting Antarctic sea ice monitoring and the evaluation of its change under global effects.

Stress distribution of a cylindrical sandwich sector with FG-CNT core and piezoelectric face sheets subjected to low-velocity impact

A new theoretical solution based on Layerwise theory is presented to determine the stress distribution in a simply-supported cylindrical sandwich sector with a functionally graded carbon nanotube core and piezoelectric face sheets subjected to low-velocity impact. The spherical elastic ball hits the top face sheet of the sector at an initial velocity of 50 m/s. The classical non-adhesive elastic contact theory and Hunter's relationship were used to determine the normal contact pressure distribution in terms of time and distance of the contact location. Moreover, Hamilton's principle and Maxwell's static equation were used to obtain the nineteen equations of motion. The numerical code was written to solve these nonlinear equations and the stress components in each layer were calculated in terms of time. In addition, it was assumed that the out-of-shell displacement of the sector at each layer is a quadratic polynomial function of the radial component in addition to a function of the coordinate components within the shell. Due to this assumption, the normal out-of-shell strain is not only zero, or not even a constant, but it changes in the form of a linear function along with the thickness of each layer.

Multi-Scale Paraxial Models to Approximate Vlasov–Maxwell Equations

Abstract Even today, solving numerically the time-dependent Vlasov–Maxwell equations is a challenging issue, and developing simpler but accurate approximate models is still worthwhile. Here, we propose a new family of paraxial asymptotic models that approximates the Vlasov–Maxwell system of equations. We introduce parameters in our models that allow us to handle relativistic cases, much slower beams or even non-relativistic cases. These models are derived by introducing a small parameter and provide static or quasi-static approximate equations that are 𝑛-th order accurate; 𝑛 may be chosen as required. Practically, one can select a model by determining the regime one is interested in and choosing the degree of accuracy needed.

Substantiation of flat blade parameters for soil loosening

Layer-by-layer loosening of soil contributes to an increase in energy consumption for its cultivation. Thus, in order to reduce traction resistance, a blade with a variable crumbling angle has been developed. In the blade stand area, it is proposed to increase crumbling angle of the chisel plow by analogy with working bodies of the chisel plow to 300, and the blade wing should be made in the form of a flat-cutting blade with a small crumbling angle. To substantiate paw profile in the strut zone, a static equation of action of forces has been drawn up depending on curve shape of the front. Distribution regularity of specific soil pressures on the blade has been determined. To verify results reliability of theoretical studies, a field experiment was carried out. Behind the tractor wheel, a beam with 4 proposed blades was installed, first row blades were set to a depth of 7 cm, and second row blades, following the trail of the first blades, to a depth of 15 cm. Using method of soil sifting on sieves with round holes, structural and aggregate composition of soil was determined along the track. Maximum lump size in the traditionally treated area is 105 mm, while maximum lump size in the layer-treated area is 80 mm, which is 24% less. Use of layer-by-layer processing in the areas compacted by the tractor propellers made it possible to improve soil pulverization quality, and thereby reduce negative factors of overcompaction by tractor running system.

Neural network guided adjoint computations in dual weighted residual error estimation

In this work, we are concerned with neural network guided goal-oriented a posteriori error estimation and adaptivity using the dual weighted residual method. The primal problem is solved using classical Galerkin finite elements. The adjoint problem is solved in strong form with a feedforward neural network using two or three hidden layers. The main objective of our approach is to explore alternatives for solving the adjoint problem with greater potential of a numerical cost reduction. The proposed algorithm is based on the general goal-oriented error estimation theorem including both linear and nonlinear stationary partial differential equations and goal functionals. Our developments are substantiated with some numerical experiments that include comparisons of neural network computed adjoints and classical finite element solutions of the adjoints. In the programming software, the open-source library deal.II is successfully coupled with LibTorch, the PyTorch C++ application programming interface.Article HighlightsAdjoint approximation with feedforward neural network in dual-weighted residual error estimation.Side-by-side comparisons for accuracy and computational cost with classical finite element computations.Numerical experiments for linear and nonlinear problems yielding excellent effectivity indices.

3D Reconstruction of Non-Lambertian Surfaces by Perspective Shape-From-Shading With Fast Viscosity Solution

Shape-from-shading (SFS) is an important method to reconstruct three-dimensional (3D) shape of a surface in photometry and computer vision. Lambertian surface reflectance and orthographic camera projection are two fundamental assumptions which generally result in undesirable reconstructed results since inaccurate imaging model is adopted. In this paper, we propose a new fast 3D shape reconstruction approach via the SFS method relaxing the two assumptions. To this end, the Oren-Nayar reflectance and perspective projection models are used to establish an image irradiance equation which depicts the relationship between the 3D shape of non-Lambertian surfaces and its corresponding two-dimensional (2D) shading image. Considering the light attenuation of the near point source, the image irradiance equation is transformed into a static Hamilton-Jacobi partial differential equation (PDE) by solving a quadratic equation. The viscosity solution of the resultant Hamilton-Jacobi PDE is approximated by using optimal control theory and iterative fast marching method starting from a viscosity supersolution. The performance of the proposed approach is evaluated on both synthetic and real-world images and the experimental results demonstrate that the proposed approach is accurate and fast.

Numerical Solutions to Wave Propagation and Heat Transfer Non-Linear PDEs by Using a Meshless Method

The applications of the Eikonal and stationary heat transfer equations in broad fields of science and engineering are the motivation to present an implementation, not only valid for structured domains but also for completely irregular domains, of the meshless Generalized Finite Difference Method (GFDM). In this paper, the fully non-linear Eikonal equation and the stationary heat transfer equation with variable thermal conductivity and source term are solved in 2D. The explicit formulae for derivatives are developed and applied to the equations in order to obtain the numerical schemes to be used. Moreover, the numerical values that approximate the functions for the considered domain are obtained. Numerous examples for both equations on irregular 2D domains are exposed to underline the effectiveness and practicality of the method.

Reduced Order AC Microgrid Model for Secondary Controller Design Using dq Reference Frame

Improving the design of primary and secondary level controllers of microgrids is of critical importance to microgrid deployment. Controller design and tuning rely heavily upon adequate, dynamic models of microgrids. In modeling of microgrids, there is always a compromise between accuracy and complexity as indicated by model order. Many of the currently available models found in the literature are reflective of system behavior but with high-order or less representative of system behavior with a low-order mathematical representation. In this paper, a reduced order, linear, inverter based, microgrid model based on the dq reference frame is developed. The order of the model is 2n (for n distributed generators (DG)) lower than the least of the orders of models suited for secondary controller design currently available. The system representation includes the commonly neglected cable impedance dynamics and the interconnection coupling between DG units. Static and dynamic model equations are utilized to capture strong correlations between the linear, reduced order (6n - 1 states for n DGs) model and non-linear system model in the transient (low frequency) and steady-state regions. The model developed is validated via a time domain simulation in the MATLAB/SIMULINK environment. The effect of the interconnection coupling between DGs on the eigenvalues is also presented.

Spontaneous non-Hermiticity in the ( 2+1 )-dimensional Gross-Neveu model

Using a nonperturbative approach based on the Cornwall-Jackiw-Tomboulis (CJT) effective action $\mathrm{\ensuremath{\Gamma}}(S)$ for composite operators ($S$ is the full fermion propagator), the phase structure of the simplest massless ($2+1$)-dimensional Gross-Neveu model is investigated. We have calculated $\mathrm{\ensuremath{\Gamma}}(S)$ and its stationary (or Dyson-Schwinger) equation in the first order of the bare coupling constant $G$ and have shown that there exist a well-defined dependence of $G\ensuremath{\equiv}G(\mathrm{\ensuremath{\Lambda}})$ on the cutoff parameter $\mathrm{\ensuremath{\Lambda}}$, such that the Dyson-Schwinger equation is renormalized. It has three different solutions for fermion propagator $S$ corresponding to possible dynamical appearance of three different mass terms in the model. One is a Hermitian, but two others are non-Hermitian and $\mathcal{P}\mathcal{T}$ even or odd. It means that two phases with spontaneous non-Hermiticity can be emerged in the system. Moreover, mass spectrum of quasiparticles is real in these non-Hermitian and $\mathcal{P}\mathcal{T}$ even/odd phases.

Multi-scale modeling and simulation for multi-pass processing of Ta-2.5 W alloy

Multi-pass deformation and annealing process is commonly utilized to prepare billets with homogeneous fine microstructure. Characterizing the deformation and microstructural evolution is not easy due to the complex process. This study proposed a multi-scale modeling and simulation method for the multi-pass processing and it was validated by experiments. Firstly, an integrate multi-scale model involving strain hardening and static softening equations was developed based on multi-pass cold deformation and annealing tests of Ta-2.5 W alloy. Of note, the model captures both deformation behavior and static softening and the model equations of each pass were integrated into a unified manner. Then, multi-scale simulation including finite element (FE) and cellular automatic (CA) methods was coupled with the developed model for static recrystallization (SRX) analysis. FE analysis was firstly performed to predict the distributions of strain, dislocation density as well as SRX. Simultaneously, the predicted data was used for generation of the initial conditions for mesoscopic model. Finally, a CA method was applied for further analysis on recrystallization nucleation and grain growth. The modeling and simulation method is examined by multi-pass cold deformation and annealing test and a reasonable consistency is obtained between experimental and predicted results.

Geodesic Rays of the N-Body Problem

For the Newtonian N-body problem, we study the Jacobi-Maupertuis metric of the nonnegative energy levels. We show that the geodesic rays are expansive, that is to say, all the distances between the bodies must be divergent functions. More precisely, we prove that the evolution of such motions asymptotically decomposes into free particles and subsystems in completely parabolic expansion. The theorem applies in particular to the maximal characteristic curves of any given global viscosity solution of the stationary Hamilton-Jacobi equation $H(x,d_xu) = h$.

An Optimization-Based Method for the Explicit Production Data Analysis of Gas Wells

This work aims at the exploration of production data analysis (PDA) methods without iterations. It can overcome limitations of the advanced type curve analysis relying on the iterative calculation of material-balance pseudotime and current explicit methods reckoning on specific production schedule assumptions. The dynamic material balance equation (DMBE) is strictly proved by the integral variable substitution based on the gas flow equation under the boundary dominated flow (BDF) condition and the static material balance equation (SMBE) of a gas reservoir. We introduce the pseudopressure level functionγ(p) and the recovery factor functionR(p) to rewrite the DMBE in terms of observed variableYand estimated variableYe; then the PDA can be transformed into an optimization problem of minimizing the error betweenYandYe. An optimization-based method for the explicit production data analysis of gas wells (OBM-EPDA), therefore, is developed in the paper, capable of determining the BDF constant and gas reserves explicitly and accurately for variable rate and/or variable flowing pressure systems. Three stimulated cases demonstrate the applicability and validity of OBM-EPDA with small errors less than 1% for estimated values of both reserves andY. Not second to previous studies, the field case analysis further proves its practicability. It is shown that the nonlinear relation ofγtoRcan be represented by a polynomial function merely dependent on the inherent properties of the gas production system even before sorting out the production data. The errors of observed variableYprovided by OBM-EPDA facilitate the data quality control, and the elimination of outliers not subject to the BDF condition improves the reliability of the analysis. For various gas systems producing whether at a constant rate, a constant bottomhole pressure (BHP), or under variable rate and variable BHP conditions, the proposed method gives insights into the well-controlled volume and production capacity of the gas well whether in a low-pressure or high-pressure gas reservoir, where the compressibilities of rock and bound water are considered.

Stationary Kirchhoff equations and systems with reaction terms

<abstract><p>In this paper, the operator approach based on the fixed point principles of Banach and Schaefer is used to establish the existence of solutions to stationary Kirchhoff equations with reaction terms. Next, for a coupled system of Kirchhoff equations, it is proved that under suitable assumptions, there exists a unique solution which is a Nash equilibrium with respect to the energy functionals associated to the equations of the system. Both global Nash equilibrium, in the whole space, and local Nash equilibrium, in balls are established. The solution is obtained by using an iterative process based on Ekeland's variational principle and whose development simulates a noncooperative game.</p></abstract>

An efficient data-driven solver for Fokker–Planck equations: algorithm and analysis

Computing the invariant probability measure of a randomly perturbed dynamical system usually means solving the stationary Fokker-Planck equation. This paper studies several key properties of a novel data-driven solver for low-dimensional Fokker-Planck equations proposed in [15]. Based on these results, we propose a new `block solver' for the stationary Fokker-Planck equation, which significantly improves the performance of the original algorithm. Some possible ways of reducing numerical artifacts caused by the block solver are discussed and tested with examples.

High Order Finite Difference Hermite WENO Fixed-Point Fast Sweeping Method for Static Hamilton-Jacobi Equations

In this paper, we combine the nonlinear HWENO reconstruction in \cite{newhwenozq} and the fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the static Hamilton-Jacobi equations in a novel HWENO framework recently developed in \cite{mehweno1}. The proposed HWENO frameworks enjoys several advantages. First, compared with the traditional HWENO framework, the proposed methods do not need to introduce additional auxiliary equations to update the derivatives of the unknown function $\phi$. They are now computed from the current value of $\phi$ and the previous spatial derivatives of $\phi$. This approach saves the computational storage and CPU time, which greatly improves the computational efficiency of the traditional HWENO scheme. In addition, compared with the traditional WENO method, reconstruction stencil of the HWENO methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping method is used to update the numerical approximation. It is an explicit method and does not involve the inverse operation of nonlinear Hamiltonian, therefore any Hamilton-Jacobi equations with complex Hamiltonian can be solved easily. It also resolves some known issues, including that the iterative number is very sensitive to the parameter $\varepsilon$ used in the nonlinear weights, as observed in previous studies. Finally, in order to further reduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate the good performance of the proposed fixed-point fast sweeping HWENO methods.

Mathematical modeling of memristor resistive switching based on mass transfer full model of oxygen vacancies and ions

A relatively simple mathematical model of dynamic switching of a memristor has been created based on a fairly complete physical model of the processes of stationary mass transfer of oxygen vacancies and ions, considering their generation, recombination and diffusion in electric field in the “metal-oxide-metal” structure with the dominant transport mechanism of electron tunneling through oxygen vacancies. The results of numerical simulation of mass transfer of oxygen vacancies along thickness of the oxide layer of the memristor are presented. The distributions of vacancy concentration during their diffusion in an electric field are found, taking into account the processes of generation and recombination with ions, depending on the applied voltage to the electrodes and on the temperature of the memristor. A good coincidence of the volt-ampere characteristics part found as a result of numerical simulation and a series of experiments is obtained. It is shown that under conditions of more than 600 K memristor temperature, it is possible to neglect the process of ion-vacancy recombination and significantly simplify the procedure for mathematical modeling of memristor resistive switching by eliminating the oxygen mass transfer equation, as well as the recombination term in the stationary equation of oxygen vacancies mass transfer. The developed mathematical model of memristor dynamic switching, including a system of stationary ordinary differential equations, is designed to simulate the operation of large memristor arrays in neuromorphic computing devices and may be preferable in relation to known circuit models that include a certain set of fitting parameters to match the simulation results with the memristor experimental characteristics.

Finite Time Stability and Optimal Finite Time Stabilization for Discrete-Time Stochastic Dynamical Systems

In this paper, we address finite time stability in probability of discrete-time stochastic dynamical systems. Specifically, a stochastic comparison lemma is constructed along with a scalar system involving a generalized deadzone function to establish almost sure convergence and finite time stability in probability. This result is used to provide Lyapunov theorems for finite time stability in probability for Itô-type stationary nonlinear stochastic difference equations involving Lyapunov difference conditions on the minimum of the Lyapunov function itself along with a fractional power of the Lyapunov function. In addition, we establish sufficient conditions for almost sure lower semicontinuity of the stochastic settling-time capturing the average settling time behavior of the discrete-time nonlinear stochastic dynamical system. Furthermore, a stochastic finite-time optimal control framework is developed by exploiting connections between Lyapunov theory for finite time stability in probability and stochastic Bellman theory. In particular, we show that finite time stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady state form of the stochastic Bellman equation, and hence, guaranteeing both stochastic finite time stability and optimality.