Full Equations Research Articles

Semi-reduced order stochastic finite element methods for solving contact problems with uncertainties

This paper develops two-step methods for solving contact problems with uncertainties. In the first step, we propose stochastic Lagrangian multiplier/penalty methods to compute a set of reduced basis. In the stochastic Lagrangian multiplier method, the stochastic solution is represented as a sum of products of a set of random variables and deterministic vectors. In the stochastic penalty method, the problem is divided into the solutions of non-contact and possible contact nodes, which are represented as sums of the products of two different sets of random variables and deterministic vectors, respectively. The original problems are then transformed into deterministic finite element equations and one-dimensional (corresponding to stochastic Lagrangian multiplier method)/two-dimensional (corresponding to stochastic penalty method) stochastic algebraic equations. The deterministic finite element equations are solved by existing numerical techniques, and the one-/two-dimensional stochastic algebraic equations are solved by a sampling method. Since the computational cost for solving stochastic algebraic equations does not increase dramatically as the stochastic dimension increases, the proposed methods avoid the curse of dimensionality in high-dimensional problems. Based on the reduced basis, we propose semi-reduced order Lagrangian multiplier/penalty equations with two components in the second step. One component is a reduced order equation obtained by smooth solutions of the reduced basis and the other is the full order equation for the nonsmooth solutions. A significant amount of computational cost is saved since the sizes of the semi-reduced order equations are usually small. Numerical examples of up to 100 dimensions demonstrate the good performance of the proposed methods.

The van der Waals Hexaquark Chemical Potential in Dense Stellar Matter

We explore the chemical potential of a QCD-motivated van der Waals (VDW) phase change model for the six-quark color-singlet, strangeness S = −2 particle known as the hexaquark with quark content (uuddss). The hexaquark may have internal structure, indicated by short range correlations that allow for non-color-singlet diquark and triquark configurations whose interactions will change the magnitude of the chemical potential. In the multicomponent VDW Equation of State (EoS), the quark-quark particle interaction terms are sensitive to the QCD color factor, causing the pairing of these terms to give different interaction strengths for their respective contributions to the chemical potential. This results in a critical temperature near 163 MeV for the color-singlet states and tens of MeV below this for various mixed diquark and triquark states. The VDW chemical potential is also sensitive to the number density, leading to chemical potential isotherms that exhibit spinodal extrema, which also depend upon the internal hexaquark configurations. These extrema determine regions of metastability for the mixed states near the critical point. We use this chemical potential with the chemical potential-modified TOV equations to investigate the properties of hexaquark formation in cold compact stellar cores in beta equilibrium. We find thresholds for hexaquark layers and changes in maximum mass values that are consistent with observations from high mass compact stellar objects such as PSR 09043 + 10 and GW 190814. In general, we find that the VDW-TOV model has an upper stability mass and radius bound for a chemical potential of 1340 MeV with a compactness of C~0.2.

Modeling of the Electronic Structure of Semiconductor Nanoparticles

This paper deals with the mathematical modeling of the electronic structure of semiconductor particles. Mathematically, the task is reduced to a joint solution of the problem of free energy minimization and the set of chemical kinetic equations describing the processes at the surface of a nanoparticle. The numerical modeling of the sensor effect is carried out in two steps. First, the number of charged oxygen atoms on the surface of the nanoparticle NO− is determined. This value is found by solving a system of nonlinear algebraic equations, where the unknowns are the stationary points of this system describing the processes on the surface of a nanoparticle. The specific form of such equations is determined by the type of nanoparticles and the mechanism of chemical reactions on the surface. The second step is to calculate the electron density inside the nanoparticle (nc(r)), which gives the minimum free energy. Mathematically, this second step reduces to solving a boundary value problem for a nonlinear integro-differential equation. The calculation results are compared with experimental data on the sensor effect.

Corrosion prevention for PEO-coated Mg by phosphate-based sealing treatment with added cation

There is no research on sealing treatments for PEO-coated Mg with phosphate sealing solutions containing different cations. In this work, four different cations are added to the phosphate solution for sealing treatment of PEO-coated Mg. The PEO coatings sealed with solutions containing Ca2+ and Mn2+ achieve the best corrosion resistance, since their dense deposition layers can seal the micropores and resist the erosion of the corrosive medium. The physical model and chemical equation used to deposit the formation of the sealing reaction are established by the experiment. They can predict the sealing effects when adding different cations to the phosphate solution. Comparing with other researches which are largely restricted to investigating the effect of phosphate-based sealing treatment on the corrosion resistance improvement of PEO coatings, conclusions here provide better guidance for formulating sealing methods to the PEO coating.

First look at the evaluation of three-loop non-planar Feynman diagrams for Higgs plus jet production

We present new computations for Feynman integrals relevant to Higgs plus jet production at three loops, including first results for a non-planar class of integrals. The results are expressed in terms of generalised polylogarithms up to transcendental weight six. We also provide the full canonical differential equations, which allows us to make structural observations on the answer. In particular, we find a counterexample to previously conjectured adjacency relations, for a planar integral of the tennis-court type. Additionally, for a non-planar triple ladder diagram, we find two novel alphabet letters. This information may be useful for future bootstrap approaches.

Specificity of the electron energy distribution function in a low-pressure nitrogen plasma

In this work, self-consistent simulations of a glow discharge positive column in low-pressure nitrogen plasma are presented. The electron energy distribution function (EEDF) was calculated by solving the complete (i.e. nonlocal) Boltzmann kinetic equation for electrons in both the energy and coordinate (space) variables. The accuracy and range of applicability of the two limiting cases are considered when the kinetic equation is reduced to a dependence on only one variable: kinetic energy in the local approximation or total energy in the so-called totally nonlocal (Holstein–Tsendin) approximation. It is shown that the Holstein–Tsendin approximation, when electron spatial transport dominates the electron energy relaxation (i.e. the ratio between the typical plasma inhomogeneity characteristic length and the electron energy relaxation length ), can be used at low nitrogen pressures (when the parameter pR ⩽ 0.05 Torr). With a further increase in pressure, there is a large intermediate region where the EEDF is nonlocal in the sense that the full kinetic equation must be solved. So, even at pR ≈ 0.5 Torr, the EEDF in nitrogen plasma is nonlocal. Therefore, the use of a local approximation to find the EEDF, when all terms with spatial gradients and an ambipolar field can be discarded in the kinetic equation, requires careful verification. Particular attention should be paid to the periphery of the discharge volume, where the self-consistent ambipolar field inevitably exceeds the external electric field that heats the electrons and produces plasma.

Simulation analysis on microscopic discharge characteristics of the bipolar corona of a floating conductor

A floating conductor exhibits a bipolar corona phenomenon with microscopic discharge characteristics that are still unclear. In this study, a plasma simulation model of the bipolar corona with 108 chemical reaction equations is established by combining hydrodynamics and plasma chemical reactions. The evolution characteristics of electrons, positive ions, negative ions and neutral particles, as well as the distribution characteristics of space charges are analyzed, and the evolutionary flow of microscopic particles is summarized. The results indicate that the positive end of the bipolar corona initiates discharge before the negative end, but the plasma chemistry at the negative end is more vigorous. The electron generation rate can reach 1240 mol (m3 s)−1, and the dissipation rate can reach 34 mol (m3 s)−1. The positive ion swarm is dominated by O+ 4, and the maximum generation rate can reach 440 mol (m3 s)−1. The negative ion swarm is mainly O− 2 and O− 4. The O− 2 content is approximately 1.5–3 times that of O− 4, and the maximum reaction rate can reach 51 mol (m3 s)−1. The final destination of neutral particles is an accumulation in the form of O3 and NO, and the amount of O3 produced is approximately 4–6 times that of NO. The positive end of the bipolar corona is dominated by positive space charges, which continue to develop and spread outwards in the form of a pulse wave. The negative end exhibits a space charge distribution structure of concentrated positive charges and diffused negative charges. The validity of the microscopic simulation analysis is verified by the macroscopic discharge phenomenon.

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.

A discrete adjoint full potential formulation for fast aerostructural optimization in preliminary aircraft design

Preliminary aircraft design is often carried out with the help of multidisciplinary optimization processes. Because there is a strong coupling between the flow and the structure of the aircraft, and because new composite materials have a unique capability of being designed for anisotropic directional stiffness and strength, these processes must model the aeroelastic behavior of the aircraft in order to be effective. Since many design variables are involved during the preliminary design stage, the optimization problems, formulated using high or medium fidelity models, can be solved using the adjoint method. Moreover, the level of fidelity of the fluid model and the associated simulation technique must also be selected with care, as they tend to be the main contributors to the computational cost. The novel contribution of the present work is twofold. Firstly, the discretized gradients of the full potential flow equation, which is a medium-fidelity model, are derived analytically so that they can be used in adjoint optimization problems. Moreover, the full potential flow solution and the computation of the gradients are implemented in an open-source and readily available finite element code. Secondly, aerodynamic shape and aerostructural optimization calculations are carried out on example wings to demonstrate the effectiveness and the computational efficiency of the proposed method. Overall, the results show that the newly implemented discrete adjoint nonlinear potential flow formulation is able to quickly optimize both the shape and the structural parameters of a typical wing. More specifically, the twist distribution along the wingspan is adapted to reduce the induced drag, the airfoils become more supercritical so that the shock strength and the associated wave drag are reduced, and the thickness of the structural elements is tailored to the loads to reduce the internal stresses. The present methodology is therefore able to deliver results at a low computational cost which are sufficiently accurate for the early design stages. Furthermore, the results obtained using the proposed methodology could be used as a starting point for the optimization calculations performed in later design stages.

Ag-based π-complexation adsorbent for paraffin/olefin separation

ABSTRACT With the massive requisition of light olefins in dedicated areas of petrochemical industry, adsorption separation approaches are developed to meet the requirements. The literature studying long-chain olefins/alkanes is scarce, and the actual adsorption capacity of liquid-phase is rarely studied. In this paper, the adsorption process of Ag-10× zeolite for long-chain alkanes/olefins was systematically studied, the preparation conditions and operating conditions of the zeolite were investigated. The samples were characterized by XRD, BET and XPS. The results show that silver ions were uniformly dispersed on the surface of the zeolite. The actual adsorption capacity of the modified zeolite was determined in intermittent experiments by liquid phase concentration difference and pore volume correction. The experimental data were fitted and the results showed that the Langmuir isotherm equation was more appropriate for adsorption, indicating that the adsorbent surface was homogeneous. The quasi-secondary kinetic chemical equation was consistent with the adsorption of olefins by the modified 10× zeolite, indicating that the process was a chemisorption process. Experimental results shows that Ag-10× zeolite is an excellent adsorbent for separating long-chain olefins/alkanes mixed systems.

Bed Shear Stress in Experimental Flash Flood Bores over Dry Beds and over Flowing Water: A Comparison of Methods

Hydraulic parameters, including bed shear stress, are challenging to calculate for flash floods. Applying theoretical equations to strongly unsteady flows requires comprehensive and accurate flow data that are difficult to collect for natural events. To empirically evaluate the extent to which simpler calculations can reasonably predict shear stresses in rapidly changing hydrographs, and to determine how bed shear stresses differ for hydraulic bores propagating over dry beds compared to shallow water, we conducted laboratory flume experiments on idealized short-duration flash floods with bores. We compared the Saint-Venant shallow water equation, which theoretically captures the depth-averaged momentum balance of gradually varying unsteady flow to eight simpler methods, with assumptions that are not met in these flows. The Saint-Venant method predicts higher shear stresses immediately after bore arrival, but all of the simpler methods are nonlinearly correlated with the Saint-Venant method even when flow is rapidly changing. While none of the methods we evaluated should strictly apply to rapid changes in depth and velocity of bores, the correlations we found between methods just after bore arrival suggest that, for applications where shear stresses must be calculated but data are insufficient to apply the full Saint-Venant equations, simpler methods may provide meaningful shear stress constraints. Compared to flood bores propagating over a dry bed, bores propagating over shallow flowing water had steeper water surfaces but resulted in significantly lower bed shear stresses immediately after bore arrival.

Theoretical investigation of braking of tearing mode rotation by resistive walls in ITER

The locking of the 2/1 tearing mode to the resistive wall in the ITER tokamak (15 MA inductive scenario 2) is investigated theoretically using a cylindrical asymptotic matching model. The model takes into account the fact that ITER plasmas will effectively be surrounded by two walls; the inner blanket module layer with a time constant of about 23 ms, and the outer vacuum vessel with a time constant of about 380 ms. The model also takes cognizance of the fact that neither the blanket module layer nor the vacuum vessel can be accurately described as “thin” walls (in the ordinarily accepted sense). The model incorporates changes in both the plasma poloidal and the toroidal angular velocity profiles, in response to the electromagnetic braking torque that develops at the rational surface, because it turns out that neoclassical poloidal flow-damping is not strong enough to completely suppress changes in the poloidal velocity. Finally, the model accurately calculates changes in the poloidal and toroidal plasma angular velocity profiles by evolving the full angular equations of motion, taking the electromagnetic braking torque, plasma inertia, plasma viscosity, and poloidal flow-damping into account. The time required for the 2/1 tearing mode to grow from a small amplitude to a sufficient one to lock to the walls is found to be about 3.5 s. The critical full radial island width at which wall locking is triggered is found to be about 9% of the plasma minor radius.

A new purification method for 2-mercaptobenzothiazole and its application as vulcanization accelerator

In this work, a new purification method for 2-Mercaptobenzothiazole (MBT) and its application as a vulcanization accelerator was reported, and the properties of MBT samples were comprehensively investigated. According to the chemical equation of this purified reaction, the byproduct of the CO2 method was easy to treat. Results showed that the melting point and yield of experimental CAMBT samples were increased to 173.6 °C and 93.11%, respectively. The chemical structure and crystal absorption peak of all MBT samples were also measured by the FTIR and XRD analysis. Thermogravimetric analysis was used to study the residue of resin in MBT samples. Element content confirmed the existence of C, N, H, and S elements, and there were no other elements in the products. What is more important, the mechanical properties of experimental CDMBT@NR and SAMBT@NR samples were better than that of UMBT@NR. Thus, this paper offers a new purification method for MBT and investigates its application as a vulcanization accelerator.

A Robust Bubble Growth Solution Scheme for Implementation in CFD Analysis of Multiphase Flows

Although the full form of the Rayleigh–Plesset (RP) equation more accurately depicts the bubble behavior in a cavitating flow than its reduced form, it finds much less application than the latter in the computational fluid dynamic (CFD) simulation due to its high stiffness. The traditional variable time-step scheme for the full form RP equation is difficult to be integrated with the CFD program since it requires a tiny time step at the singularity point for convergence and this step size may be incompatible with time marching of conservation equations. This paper presents two stable and efficient numerical solution schemes based on the finite difference method and Euler method so that the full-form RP equation can be better accepted by the CFD program. By employing a truncation bubble radius to approximate the minimum bubble size in the collapse stage, the proposed schemes solve for the bubble radius and wall velocity in an explicit way. The proposed solution schemes are more robust for a wide range of ambient pressure profiles than the traditional schemes and avoid excessive refinement on the time step at the singularity point. Since the proposed solution scheme can calculate the effects of the second-order term, liquid viscosity, and surface tension on the bubble evolution, it provides a more accurate estimation of the wall velocity for the vaporization or condensation rate, which is widely used in the cavitation model in the CFD simulation. The legitimacy of the solution schemes is manifested by the agreement between the results from these schemes and established ones from the literature. The proposed solution schemes are more robust in face of a wide range of ambient pressure profiles.

Bilinear Optimal Control of the Keller–Segel Logistic Model in 2D-Domains

An optimal control problem associated to the Keller–Segel with logistic reaction system is studied in 2D domains. The control acts in a bilinear form only in the chemical equation. The existence of an optimal control and a necessary optimality system are deduced. The main novelty is that the control can be rather singular and the state (cell density u and the chemical concentration v) remains only in a weak setting, which is not usual in the literature.

All-Speed Numerical Methods for the Euler Equations via a Sequential Explicit Time Integration

This paper presents a new strategy to deal with the excessive diffusion that standard finite volume methods for compressible Euler equations display in the limit of low Mach number. The strategy can be understood as using centered discretizations for the acoustic part of the Euler equations and stabilizing them with a leap-frog-type (“sequential explicit”) time integration, a fully explicit method. This time integration takes inspiration from time-explicit staggered grid numerical methods. In this way, advantages of staggered methods carry over to collocated methods. The paper provides a number of new collocated schemes for linear acoustic/Maxwell equations that are inspired by the Yee scheme. They are then extended to an all-speed method for the full Euler equations on Cartesian grids. By taking the opposite view and taking inspiration from collocated methods, the paper also suggests a new way of staggering the variables which increases the stability as compared to the traditional Yee scheme.

A Novel Hydro-Thermal Synthesis of Nano-Structured Molybdenum-Iron Intermetallic Alloys at Relatively Low Temperatures

Nano-structured Mo/Fe intermetallics were synthesized from precursors that contained 72/28% and 30/70% molar ratios of Mo/Fe, which were given as precursors A and B, respectively. These precursors were prepared from the co-precipitation of aqueous hot solutions of ammonium heptamolybdate tetrahydrate (AHM) and ferrous oxalate. The dry precipitates were thermally treated using TG-DSC to follow up their behavior during roasting, in an Ar atmosphere of up to 700 °C (10° K/min). The TG profile showed that 32.5% and 55.5% weight losses were measured from the thermal treatment of precursors A and B, respectively. The DSC heat flow profile showed the presence of endothermic peaks at 196.9 and 392.5-400 °C during the thermal decomposition of the AHM and ferrous oxalate, respectively. The exothermic peak that was detected at 427.5 °C was due to the production of nano-sized iron molybdate [Fe2(MoO4)3]. An XRD phase analysis indicated that iron molybdate was the only phase that was identified in precursor A, while iron molybdate and Fe2O3 were produced in precursor B. Compacts were made from the pressing of the nano-sized precursors, which were roasted at 500 °C for 3 h. The roasted compacts were isothermally reduced in H2 at 600-850 °C using microbalance, and the O2 weight loss that resulted from the reduction reactions was continuously recorded as a function of time. The influence of the reduction temperature and precursor composition on the reduction behavior of the precursors was studied and discussed. The partially and completely reduced compacts were examined with X-ray powder diffraction (XRD), a reflected light microscope (RLM), and a scanning electron microscope (SEM-EDS). Depending on the precursor composition, the reduction reactions of the [Fe2(MoO4)3] and Fe2O3 proceeded through the formation of intermediate lower oxides, prior to the production of the MO/Fe intermetallic alloys. Based on the intermediate phases that were identified and characterized at the early, intermediate, and final reduction degrees, chemical reaction equations were given to follow up the formation of the MoFe and MoFe3 intermetallic alloys. The mechanism of the reduction reactions was predicted from the apparent activation energy values (Ea) that were computed at the different reduction degrees. Moreover, mathematical formulations that were derived from the gas-solid reaction model were applied to confirm the reduction mechanisms, which were greatly dependent on the precursor composition and reduction temperature. However, it can be reported that nano-structured MoFe and MoFe3 intermetallic alloys can be successfully fabricated via a gas-solid reaction technique at lower temperatures.

The characteristic initial value problem for the conformally invariant wave equation on a Schwarzschild background

We resume former discussions of the conformally invariant wave equation on a Schwarzschild background, with a particular focus on the behaviour of solutions near the ‘cylinder’, i.e. Friedrich’s representation of spacelike infinity. This analysis can be considered a toy model for the behaviour of the full Einstein equations and the resulting logarithmic singularities that appear to be characteristic for massive spacetimes. The investigation of the Cauchy problem for the conformally invariant wave equation (Frauendiener and Hennig 2018 Class. Quantum Grav. 35 065015) showed that solutions generically develop logarithmic singularities at infinitely many expansion orders at the cylinder, but an arbitrary finite number of these singularities can be removed by appropriately restricting the initial data prescribed at t = 0. From a physical point of view, any data at t = 0 are determined from the earlier history of the system and hence not exactly ‘free data’. Therefore, it is appropriate to ask what happens if we ‘go further back in time’ and prescribe initial data as early as possible, namely at a portion of past null infinity, and on a second past null hypersurface to complete the initial value problem. Will regular data at past null infinity automatically lead to a regular evolution up to future null infinity? Or does past regularity restrict the solutions too much, and regularity at both null infinities is mutually exclusive? Or do we still have suitable degrees of freedom for the data that can be chosen to influence regularity of the solutions to any desired degree? In order to answer these questions, we study the corresponding characteristic initial value problem. In particular, we investigate in detail the appearance of singularities at expansion orders for angular modes .

Density and Small Disturbance Equation-Based Simulation of Designed Bipalne Airfoil Under Supersonic Flow

For inviscid airflows, the small disturbance equation (SDE), as a condensed form of the full potential equation, is an effective approach for characterising the behaviour of this type of airflow in the presence of thin obstacles in its path. Density-based solutions (DBS) are frequently used by ANSYS to model the airflow behavior under the same environment. Through the impact of shock interference between the target wings, special Bussemann-type supersonic biplane designs can be effective in restricting sonic booms and wave drags in particular supersonic conditions. The modeling investigations of the shock oscillations produced between the wings are quite rare, nevertheless. Therefore, in order to replicate the behavior of the airflow over the target wing in supersonic settings and to compare the results more closely to reality, the Small Disturbance Equation (SDE) and the Density Based Solution (DBS) will be employed as efficient simulation tools in this study. The SDE results are vulnerable to inaccuracies because of the constraints of the simulation settings, whereas the DBS results are more accurate when compared to the exact simulation results. Overall, the DBS simulation results and the legit results are consistent, indicating that the inviscid airflow hypothesis is a feasible option for modelling the effectiveness of shock oscillations between wings. The SDE approach, which is implemented in Python and is a condensed version of the en-tire potential equation, offers a lot of space for development as it only does simulations in terms of velocity in this article and ignores the pressure and density elements.

The Influence of Fluid Dynamics on the Design of Different Vehicle Structures

It is of great significance to understand the airflow disturbance above the vehicle for the design of the vehicle model. However, the traditional method is wind tunnel test, which is inconvenient and time wasted. In this article the team simulated the curves of different shapes above cars, such as Beetle, modern sedan car, jeep, and wedge-shaped cars. Besides, calculated the small disturbance equation in the form of python code. This paper anaylised the full potential equation to find the small disturbance equation. Then studied and analyzed the disturbance of subsonic airflow above cars. Finally, this paper analysed the functional image and did a conclusion. The results shows that different shape of cars has different flow above the car in the subsonic flow and the wedge car has almost the best shape. The experiment result shows that it has an exelent accuracy compared to the literature, besided it has a low cost and a shorter time waste.