Terms Of Relaxation Research Articles

A Novel k-Means Framework via Constrained Relaxation and Spectral Rotation.

Owing to its simplicity, the traditional k - means (Lloyd heuristic) clustering method plays a vital role in a variety of machine-learning applications. Disappointingly, the Lloyd heuristic is prone to local minima. In this article, we propose k - mRSR, which converts the sum-of-squared error (SSE) (Lloyd) into a combinatorial optimization problem and incorporates a relaxed trace maximization term and an improved spectral rotation term. The main advantage of k - mRSR is that it only needs to solve the membership matrix instead of computing the cluster centers in each iteration. Furthermore, we present a nonredundant coordinate descent method that brings the discrete solution infinitely close to the scaled partition matrix. Two novel findings from the experiments are that k - mRSR can further decrease (increase) the objective function values of the k - means obtained by Lloyd (CD), while Lloyd (CD) cannot decrease (increase) the objective function obtained by k - mRSR. In addition, the results of extensive experiments on 15 datasets indicate that k - mRSR outperforms both Lloyd and CD in terms of the objective function value and outperforms other state-of-the-art methods in terms of clustering performance.

Semi-implicit-Type Order-Adaptive CAT2 Schemes for Systems of Balance Laws with Relaxed Source Term

AbstractIn this paper, we present two semi-implicit-type second-order compact approximate Taylor (CAT2) numerical schemes and blend them with a local a posteriori multi-dimensional optimal order detection (MOOD) paradigm to solve hyperbolic systems of balance laws with relaxed source terms. The resulting scheme presents the high accuracy when applied to smooth solutions, essentially non-oscillatory behavior for irregular ones, and offers a nearly fail-safe property in terms of ensuring the positivity. The numerical results obtained from a variety of test cases, including smooth and non-smooth well-prepared and unprepared initial conditions, assessing the appropriate behavior of the semi-implicit-type second order CATMOOD schemes. These results have been compared in the accuracy and the efficiency with a second-order semi-implicit Runge-Kutta (RK) method.

A modified inertial proximal alternating direction method of multipliers with dual-relaxed term for structured nonconvex and nonsmooth problem

In this research, we introduce a novel optimization algorithm termed the dual-relaxed inertial alternating direction method of multipliers (DR-IADM), tailored for handling nonconvex and nonsmooth problems. These problems are characterized by an objective function that is a composite of three elements: a smooth composite function combined with a linear operator, a nonsmooth function, and a mixed function of two variables. To facilitate the iterative process, we adopt a straightforward parameter selection approach, integrate inertial components within each subproblem, and introduce two relaxed terms to refine the dual variable update step. Within a set of reasonable assumptions, we establish the boundedness of the sequence generated by our DR-IADM algorithm. Furthermore, leveraging the Kurdyka–Łojasiewicz (KŁ) property, we demonstrate the global convergence of the proposed method. To validate the practicality and efficacy of our algorithm, we present numerical experiments that corroborate its performance. In summary, our contribution lies in proposing DR-IADM for a specific class of optimization problems, proving its convergence properties, and supporting the theoretical claims with numerical evidence.

Applying satellite sea surface temperature as Dirichlet-type surface thermal boundary condition in an ocean model

Following the Dirichlet-type boundary condition that specifies value at boundary of a system in mathematics and physics, this study suggests applying satellite sea surface temperature (SST) as Dirichlet-type surface thermal boundary condition (STBC) in ocean models. Numerical experiments with Dirichlet-type and Combined-type STBC with different configurations of satellite SST, for the period of January to April 2019 in the Northwest Pacific Ocean, were carried out based on the Princeton Ocean Model. The experiments were assessed using satellite and in situ observations of temperature, GOFS3.1 analysis and GLORYS12v1 reanalysis. Results show that applying satellite SST as Dirichlet-type STBC could constrain the modeled SST and near-surface temperature in upper 50 m depth well, which is better than the scheme that uses satellite SST as the relaxation terms of Combined-type STBC. The temperature section along the 137°E resulted from the Dirichlet-type STBC is comparable with GOFS3.1 analysis and better than GLORYS12v1 reanalysis in upper 50 m depth. These results suggest that applying high-accuracy satellite SST as Dirichlet-type STBC in ocean models has a promising prospect in numerical simulation.

Progress in electron energy modeling for plasma flows and discharges

A novel formulation of the electron energy relaxation terms is presented here, which is applicable to plasma flows and discharges wherein the electron temperature could be higher or lower than the gas temperature. It is demonstrated that the electron energy losses due to inelastic collisions can be expressed as a function of only two species-dependent parameters: the reduced electric field and the reduced electron mobility. This formulation is advantageous over previous ones, being simpler to implement and more accurate when experimental data of the reduced electric field and reduced mobility are available. Curve fits to empirical data of these two properties are outlined here for all important air molecular species. The approach accounts for all inelastic electron energy relaxation processes without needing individual cross sections or rates, reducing potential errors associated with independently handling each process. Several test cases are presented to validate the proposed electron energy source terms including reentry plasma flows for which the electron temperature is less than the gas temperature, as well as discharges in which the electron temperature reaches values in excess of 30 eV. In all cases, the agreement with experimental data is observed to be very good to excellent, significantly surpassing prior electron energy models for plasma flows.

Numerical Solution of the Newtonian Plane Couette Flow with Linear Dynamic Wall Slip

An efficient numerical approach based on weighted-average finite differences is used to solve the Newtonian plane Couette flow with wall slip, obeying a dynamic slip law that generalizes the Navier slip law with the inclusion of a relaxation term. Slip is exhibited only along the fixed lower plate, and the motion is triggered by the motion of the upper plate. Three different cases are considered for the motion of the moving plate, i.e., constant speed, oscillating speed, and a single-period sinusoidal speed. The velocity and the volumetric flow rate are calculated in all cases and comparisons are made with the results of other methods and available results in the literature. The numerical outcomes confirm the damping with time and the lagging effects arising from the Navier and dynamic wall slip conditions and demonstrate the hysteretic behavior of the slip velocity in following the harmonic boundary motion.

Revisiting the Becker-Kistiakowsky-Wilson equation of state

The Jones-Wilkins-Lee (JWL) equation of state (EOS) is the most popular equation of state used in hydrocodes to model detonation products thermodynamics resulting from high explosives. However, the JWL EOS presents several major difficulties. Namely, its range of validity and convexity is limited as its parameters are adjusted on a given reference curve. When used in conditions where the thermodynamic state varies significantly, computational failure may happen. It occurs frequently in multiphase flow computations when heat and mass transfers with other phases are present, as the resulting thermodynamic state is far from the reference curve used for adjustment. Moreover, the detonation products composition is absent in the JWL formulation. This is another important limitation when additional physics are involved, such as post-combustion, phase transition, or when dealing with non-ideal explosives. In this work, the Becker-Kistiakowsky-Wilson (BKW) EOS is considered instead. The BKW EOS is widely used in thermochemical codes, as parent EOS to provide the reference data used to fit JWL EOS parameters. The BKW EOS present several advantages. Its formulation considers variable gas species in the detonation products. No specific adjustment is needed for a given explosive, and only the detonation products composition requires computational efforts. Its range of validity is large, as the various parameters have been adjusted for each gas species separately. However, the BKW EOS also present difficulties which limit its application in hydrocodes. These difficulties are the aim of the present work. The BKW EOS is currently restricted to thermochemical codes as its mathematical formulation is computationally expensive. It also requires the gas chemical composition to be computed via an appropriate chemical equilibrium solver. In the present work, the computational cost of the BKW EOS is reduced thanks to the thermodynamic relaxation method of Neron et al. (2023). The variable composition of the detonation products is accounted for through specific relaxation terms computed once, with the help of conventional thermochemical codes. The condensed species, such as solid carbon, are modelled with the Cochran-Chan (CC) EOS, having a simpler thermodynamic formulation and being thermodynamically consistent compared to usual EOSs used for condensed species. This multiphase formulation made of BKW gas mixture with variable composition and condensed phases is considered in temperature and pressure equilibrium. It is embedded in a multiphase flow formulation, with the help of the thermodynamic relaxation method mentioned above. It shows comparable computational cost as the JWL formulation, with significantly enhanced physics capabilities and improved robustness.

Method to quickly estimate T1 value by suppressing spin exchange relaxation and magnetic field gradient relaxation in atomic sensors

The accuracy of atomic sensor is determined by the transverse nuclear spin relaxation time (T2). As the longitudinal nuclear spin relaxation time (T1) is the upper limit of T2, it becomes an important metric for evaluating the merits of alkali vapor cells. However, the conventional method for measuring T1 of pulse delay takes dozens of minutes, which will increase the effect of random errors during the testing process. In order to solve this problem, a method to quickly estimate the value of T1 is proposed by suppressing the influence of spin exchange relaxation and magnetic field gradient relaxation terms on T2. Experimental results indicate that the accuracy is maintained at above 95%, and the testing time is reduced to less than 5%.

A relaxation approach to modeling properties of hyperbolic–parabolic type models

In this work, we propose a novel relaxation modeling approach for partial differential equations (PDEs) involving convective and diffusive terms. We reformulate the original convection–diffusion problem as a system of hyperbolic equations coupled with relaxation terms. In contrast to existing literature on relaxation modeling, where the solution of the reformulated problem converges to certain types of equations in the diffusive limit, our formalism treats the augmented problem as a system of coupled hyperbolic equations with relaxation acting on both the convective flux and the source term. Furthermore, we demonstrate that the new system of equations satisfies Liu’s sub-characteristic condition. To verify the robustness of our proposed approach, we perform numerical experiments on various important models, including nonlinear convection–diffusion problems with discontinuous coefficients. The results show the promising potential of our relaxation modeling approach for both pure and applied mathematical sciences, with applications in different models and areas.

New-Generation Electron-Propagator Methods for Molecular Electron-Binding Energies.

A new generation of electron-propagator methods for the calculation of electron binding energies has surpassed its antecedents with respect to accuracy, efficiency, and interpretability. No adjustable parameters are introduced in these fully ab initio procedures. Numerical tests versus several databases of valence, vertical electron binding energies of closed-shell molecules and atoms have been performed. Easily interpreted self-energy approximations with cubic arithmetic scaling produce mean absolute errors (MAEs) of 0.2 and 0.3 eV for electron detachments and attachments, respectively. The most accurate explicitly renormalized methods with fifth-power arithmetic scaling yield MAEs below 0.1 eV for detachments and attachments. Approximate renormalization leads to more efficient fifth-power alternatives for electron detachments that achieve similar accuracy with fewer bottleneck operations. Composite protocols generate excellent predictions versus highly accurate basis-extrapolated standards and experiments. The validity of the diagonal self-energy approximation and the accuracy of the approximate renormalizations are confirmed. The success of these perturbative methods based on canonical Hartree-Fock orbitals rests on a Hermitized, intermediately normalized superoperator metric. The results of all of the new-generation calculations may be analyzed in terms of final-state orbital relaxation and differential correlation effects.

A novel and sustainable rubber composite prepared from electric arc furnace slag as carbon black replacement

Carbon black (CB) is the most widely used reinforcing filler for rubber. Nowadays there are several concerns regarding this traditional petroleum-based filler: on one side its environmental footprint is enormous and its production process is no more sustainable and on the other side its price increases annually. For these reasons, sustainable alternative fillers are being studied. In the present research the main waste of the steel industry, namely the steel slag from electric arc furnace (EAF), is investigated as non-conventional filler for a nitrile butadiene rubber matrix (NBR). The slag has been characterized to ensure its safe reuse as filler according to the heavy metals leaching. The slag filled compounds have been characterized and compared to CB filled compounds, in terms of processability by rheometric parameters, mechanical properties, Payne effect, and physicochemical properties to investigate the filler-matrix interaction. From the obtained results, it was shown that EAF slag-filled NBRs are comparable to CB filled NBRs in terms of crosslink kinetics and, when compared at the same hardness level, are comparable in terms of viscosity, stiffness, and elongation at break, while when compared at the same filler volume fraction are similar in terms of compression set and stress relaxation.

Relaxation terms for anomalous hydrodynamic transport in Weyl semimetals from kinetic theory

We consider as a model of Weyl semimetal thermoelectric transport a (3 + 1)-dimensional charged, relativistic and relaxed fluid with a U(1)V × U(1)A chiral anomaly. We take into account all possible mixed energy, momentum, electric and chiral charge relaxations, and discover which are compatible with electric charge conservation, Onsager reciprocity and a finite DC conductivity. We find that all relaxations respecting these constraints necessarily render the system open and violate the second law of thermodynamics. We then demonstrate how the relaxations we have found arise from kinetic theory and a modified relaxation time approximation. Our results lead to DC conductivities that differ from those found in the literature opening the path to experimental verification.

Understanding Complex Traffic Dynamics with the Nondimensionalisation Technique

Hydrodynamic traffic models are crucial to optimizing transportation efficiency and urban planning. They usually comprise a set of coupled partial differential equations featuring an arbitrary number of terms that aim to describe the different nuances of traffic flow. Consequently, traffic models quickly become complicated to solve and difficult to interpret. In this article, we present a general traffic model that includes a relaxation term and an inflow of vehicles term and utilize the mathematical technique of nondimensionalisation to obtain universal solutions to the model. Thus, we are able to show extreme sensitivity to initial conditions and parameter changes, a classical signature of deterministic chaos. Moreover, we obtain simple relations among the different variables governing traffic, thus managing to efficiently describe the onset of traffic jams. We validate our model by comparing different scenarios and highlighting the model’s applicability regimes in traffic equations. We show that extreme speed values, or heavy traffic inflow, lead to divergences in the model, showing its limitations but also demonstrating how the problem of traffic jams can be alleviated. Our results pave the way to simulating and predicting traffic accurately on a real-time basis.

Modeling and simulation of the cavitation phenomenon in turbopumps

This paper presents a model for simulating a two-phase flow involving cavitation and condensation, focusing on its application to turbopump flows. The model is derived by extending Kapila's approach, which incorporates thermal and free energy relaxation terms, to compressible multiphase flow equations in a rotating reference frame. The numerical method employed is a finite volume approach based on the Godunov scheme, ensuring pressure equilibrium and total energy conservation through a relaxation step. The model utilizes separate Equations of State (EOS) for the liquid and vapor phases, with the experimental saturation curve, expressed as psat(T), being the only parameter needed for cavitation modeling. To broaden its applicability beyond axial pumps, a specific Riemann solver is developed to couple rotating and non-rotating regions within the flow domain. The thermodynamic phase change process is validated using simple cases, and the model is further utilized to simulate water flows in a turbopump inducer. The characteristic curve of the pump is obtained in the non-cavitating regime for several mass flow rates conditions, and the model is successfully assessed in severe cavitation conditions, demonstrating the performance breakdown caused by vapor pockets. The modeling also provides detailed flow descriptions, including analysis of vapor pockets and their impact on blade pressure loading.

Stabilisation of stochastic single-file dynamics using port-Hamiltonian systems

This study revisits a recently proposed symmetric port-Hamiltonian single-file model in one dimension. The uniform streaming solutions are stable in the deterministic model. However, the introduction of white noise into the dynamics causes the model to exhibit divergence. In response, we add a relaxation term that draws the agents’ speed to a desired constant speed and plays the role of the input in the port-Hamiltonian framework. Our results show that this relaxation term effectively stabilises the dynamics even in the presence of stochastic noise.

Advancing the temporal direct deconvolution method with spatial regularization

This study continues the exploration of temporal large-eddy simulation, particularly the extension of the temporal direct deconvolution method (TDDM) with a regularization term based on spatial dissipation. Furthermore, we aim to put insight stemming from previous work to test. Specifically, the hypothesis is that the temporal residual-stress leads to a reduction of the required artificial dissipation in under-resolved simulations. Moreover, this work seeks corroborate earlier discoveries with a posteriori results. We perform a numerical examination of two different spatial regularization terms in conjunction with TDDM: a spatial variant of selective frequency damping, functioning as a relaxation term that gradually drifts the velocity toward the filtered velocity, and the dynamic Smagorinsky model incorporating a prefactor. We test various cases, including the Taylor–Green vortex flow with a Reynolds number of Re = 3000, forced homogeneous isotropic turbulence with Reλ=200, turbulent channel flow at Reτ=590, and the flow over a periodic hill with Re = 10 935. Additionally, we also analyze the various dissipation contributions in TDDM as well as their interrelations. We also discuss grid artifacts and energy budget errors using these to compare the different models. Our results confirm the hypothesis that residual-stress dissipation reduces the necessary artificial dissipation. Because of the numerical ill-conditioning of deconvolution, whether temporal or spatial, there are practical limitations in the size of the filter width. Due to these limitations, the impact remains relatively minor. The a posteriori results of the new spatial regularization term show it to be effective in eliminating energy from the high wavenumber range.

Validation of the Indonesian version of the Recovery Experience Questionnaire.

This study aimed to validate the Indonesian version of the Recovery Experience Questionnaire (REQ-I) to assess how individuals unwind and recuperate from work during their off-job times, specifically in terms of psychological detachment, relaxation, mastery, and control. The translated and back-translated REQ, which has gone through semantic and face validation processes, was administered through an internet survey using 740 Indonesian workers from various backgrounds. Furthermore, confirmatory Factor Analysis (CFA) was conducted to evaluate factorial validity. Construct validity was evaluated based on the correlation coefficients between recovery experience and potential consequences variables, while internal consistency and test-retest reliability were investigated to evaluate the reliability. The result of CFA showed that the hypothesized four-factor model was the most suitable for the data. Meanwhile, construct validity was supported by expected correlations of recovery experiences with possible consequences. Cronbach's α coefficient for each of the four subscales was sufficient at 0.85-0.92. Test-retest reliability of each of the four subscales with three months intervals was confirmed with sufficient intraclass correlation coefficients at 0.60-0.66. This current study confirmed that REQ-I was an adequate measure of recovery experiences used in the Indonesian context.

High-order relaxation methods for nonequilibrium two-phase flow equations

PurposeThe purpose of this paper is to investigate the two-phase flow problems involving gas–liquid mixture.Design/methodology/approachThe governed equations consist of a range of conservation laws modeling a classification of two-phase flow phenomena subjected to a velocity nonequilibrium for the gas–liquid mixture. Effects of the relative velocity are accounted for in the present model by a kinetic constitutive relation coupled to a collection of specific equations governing mass and volume fractions for the gas phase. Unlike many two-phase models, the considered system is fully hyperbolic and fully conservative. The suggested relaxation approach switches a nonlinear hyperbolic system into a semilinear model that includes a source relaxation term and characteristic linear properties. Notably, this model can be solved numerically without the use of Riemann solvers or linear iterations. For accurate time integration, a high-resolution spatial reconstruction and a Runge–Kutta scheme with decreasing total variation are used to discretize the relaxation system.FindingsThe method is used in addressing various nonequilibrium two-phase flow problems, accompanied by a comparative study of different reconstructions. The numerical results demonstrate the suggested relaxation method’s high-resolution capabilities, affirming its proficiency in delivering accurate simulations for flow regimes characterized by strong shocks.Originality/valueWhile relaxation methods exhibit notable performance and competitive features, as far as we are aware, there has been no endeavor to address nonequilibrium two-phase flow problems using these methods.

Numerical and Theoretical Analysis of Model Equations for Multicomponent Rarefied Gas

Model equations approximating the system of Boltzmann equations for a multicomponent gas are investigated. Methods for determining parameters in relaxation terms corresponding to cross-collision integrals are analyzed. Numerical solutions based on three model systems and the Boltzmann equations are compared as applied to the following problems: relaxation of a mixture to equilibrium, shock wave structure, and the dynamics of a vapor-gas cloud generated by pulsed laser irradiation of a target. It is shown that the parameters in the relaxation operators influence the degree of difference in the solutions produced by the various models.

Characterizing linearizable QAPs by the level-1 reformulation-linearization technique

The quadratic assignment problem (QAP) is an extremely challenging NP-hard combinatorial optimization program. Due to its difficulty, a research emphasis has been to identify special cases that are polynomially solvable. Included within this emphasis are instances which are linearizable; that is, which can be rewritten as a linear assignment problem having the property that the objective function value is preserved at all feasible solutions. Various known sufficient conditions for identifying linearizable instances have been explained in terms of the continuous relaxation of a weakened version of the level-1 reformulation-linearization-technique (RLT) form that does not enforce nonnegativity on a subset of the variables. Also, conditions that are both necessary and sufficient have been given in terms of decompositions of the objective coefficients. The main contribution of this paper is the identification of a relationship between polyhedral theory and linearizability that promotes a novel, yet strikingly simple, necessary and sufficient condition for identifying linearizable instances; specifically, an instance of the QAP is linearizable if and only if the continuous relaxation of the same weakened RLT form is bounded. In addition to providing a novel perspective on the QAP being linearizable, a consequence of this study is that every linearizable instance has an optimal solution to the (polynomially-sized) continuous relaxation of the level-1 RLT form that is binary. The converse, however, is not true so that the continuous relaxation can yield binary optimal solutions to instances of the QAP that are not linearizable. Another consequence follows from our defining a maximal linearly independent set of equations in the lifted RLT variable space; we answer a recent open question that the theoretically best possible linearization-based bound cannot improve upon the level-1 RLT form.