Data-driven optimization study of the multi-relaxation-time lattice Boltzmann method for solid-liquid phase change

Sharp phase interfaces and accurate temperature distributions are important criteria in the simulation of solid-liquid phase changes.The multi-relaxation-time lattice Boltzmann method(MRT-LBM)shows great numerical performance during simulation;however,the value method of the relaxation parameters needs to be specified.Therefore,in this study,a random forest(RF)model is used to discriminate the importance of different relaxation parameters to the convergence,and a support vector machine(SVM)is used to explore the decision boundary of the convergent samples in each dimensional model.The results show that the convergence of the samples is consistent with the sign of the decision number,and two types of the numerical deviations appear,i.e.,the phase mushy zone and the non-physical heat transfer.The relaxation parameters chosen on the decision boundary can further suppress the numerical bias and improve numerical accuracy.

GENERALIZED MATRIX MULTISPLITTING RELAXATION METHODS AND THEIR CONVERGENCE

In this paper, we set up a general framework of parallel matrix mullisplitting relaxation methods for solving large scale system of linear equations. We investigate the convergence properties of this framework and give several sufficient conditions ensuring it to converge as well as diverge. At last, we conclude a necessary and sufficient condition for the convergence of this framework when the coefficient matrix is an L-matrix.

A time domain induced polarization relaxation time spectrum inversion method based on a damping factor and residual correction

Relaxation time spectra （RTS） derived from time domain induced polarization data （TDIP） are helpful to assess oil reservoir pore structures. However, due to the sensitivity to the signal-to-noise ratio （SNR）, the inversion accuracy of the traditional singular value decomposition （SVD） inversion method reduces with a decrease of SNR. In order to enhance the inversion accuracy and improve robustness of the inversion method to the SNR, an improved inversion method, based on damping factor and spectrum component residual correction, is proposed in this study. The numerical inversion results show that the oscillation of the RTS derived from the SVD method increased with a decrease of SNR, which makes it impossible to get accurate inversion components. However, the SNR has little influence on inversion components of the improved method, and the RTS has high inversion accuracy and robustness. Moreover, RTS derived from core sample data is basically in accord with the pore-size distribution curve, and the RTS derived from the actual induced polarization logging data is smooth and continuous, which indicates that the improved method is practicable.

Geometric interpretation of several classical iterative methods for linear system of equations and diverse relaxation parameter of the SOR method

Two kinds of iterative methods are designed to solve the linear system of equations, we obtain a new interpretation in terms of a geometric concept. Therefore, we have a better insight into the essence of the iterative methods and provide a reference for further study and design. Finally, a new iterative method is designed named as the diverse relaxation parameter of the SOR method which, in particular, demonstrates the geometric characteristics. Many examples prove that the method is quite effective.

Combined immersed boundary method and multiple-relaxation-time lattice Boltzmann flux solver for numerical simulations of incompressible flows

A method combining the immersed boundary technique and a multi- relaxation-time （MRT） lattice Boltzmann flux solver （LBFS） is presented for numerical simulation of incompressible flows over circular and elliptic cylinders and NACA 0012 Airfoil. The method uses a simple Cartesian mesh to simulate flows past immersed complicated bodies. With the Chapman-Enskog expansion analysis, a transform is performed between the Navier-Stokes and lattice Boltzmann equations （LBEs）. The LBFS is used to discretize the macroscopic differential equations with a finite volume method and evaluate the interface fluxes through local reconstruction of the lattice Boltzmann solution. The immersed boundary technique is used to correct the intermediate velocity around the solid boundary to satisfy the no-slip boundary condition. Agreement of simulation results with the data found in the literature shows reliability of the proposed method in simulating laminar flows on a Cartesian mesh.

Modified Exact Jacobian Semidefinite Programming Relaxation for Celis-Dennis-Tapia Problem

A modified exact Jacobian semidefinite programming(SDP)relaxation method is proposed in this paper to solve the Celis-Dennis-Tapia(CDT)problem using the Jacobian matrix of objective and constraining polynomials.In the modified relaxation problem,the number of introduced constraints and the lowest relaxation order decreases significantly.At the same time,the finite convergence property is guaranteed.In addition,the proposed method can be applied to the quadratically constrained problem with two quadratic constraints.Moreover,the efficiency of the proposed method is verified by numerical experiments.

A multiple-relaxation-time lattice Boltzmann method for high-speed compressible flows

This paper presents a coupling compressible model of the lattice Boltzmann method. In this model, the multiplerelaxation-time lattice Boltzmann scheme is used for the evolution of density distribution functions, whereas the modified single-relaxation-time （SRT） lattice Boltzmann scheme is applied for the evolution of potential energy distribution functions. The governing equations are discretized with the third-order Monotone Upwind Schemes for scalar conservation laws finite volume scheme. The choice of relaxation coefficients is discussed simply. Through the numerical simulations, it is found that compressible flows with strong shocks can be well simulated by present model. The numerical results agree well with the reference results and are better than that of the SRT version.

Simulating high Reynolds number flow in two-dimensional lid-driven cavity by multi-relaxation-time lattice Boltzmann method

By coupling the non-equilibrium extrapolation scheme for boundary condition with the multi-relaxation-time lattice Boltzmann method, this paper finds that the stability of the multi-relaxation-time model can be improved greatly, especially on simulating high Reynolds number （Re） flow. As a discovery, the super-stability analysed by Lallemand and Luo is verified and the complex structure of the cavity flow is also exhibited in our numerical simulation when Re is high enough. To the best knowledge of the authors, the maximum of Re which has been investigated by direct numerical simulation is only around 50 000 in the literature; however, this paper can readily extend the maximum to 1000 000 with the above combination.

PARALLEL NONLINEAR MULTISPLITTING RELAXATION METHODS

By further generalizing Frommer's results in the sense of nonlinear multisplitting, we build a class of nonlinear multisplitting AOR-type methods, which covers many rather practical nonlinear multisplitting relaxation methods such as multisplitting AOR-Newton method, multisplitting AOR-chord method and multisplitting AOR-Steffensen method, etc.. Furthermore,a general convergence theorem for the nonlinear multisplitting AOR-type methods and the local convergence for the multisplitting AOR-Newton method are discussed in detail.A lot of numerical tests show that our new methods are feasible and satisfactory.

Hydrodynamic Lubrication of Elastic Foil Gas Bearing Using Over Relaxation Iteration Method and Non-Dimensional Equation

The purpose is to accurately predict the performance of foil bearing and achieve accurate results in the design of foil bearing structure.A new type of foil bearing with surface microstructure is used as experimental material.First,the lubrication mechanism of elastic foil gas bearing is analyzed.Then,the numerical solution process of the static bearing capacity and friction torque is analyzed,including the discretization of the governing equation of rarefied gas pressure based on the non-dimensional modified Reynolds equation and the over relaxation iteration method,the grid planning within the calculation range,the static solution of boundary parameters and static solution of the numerical process.Finally,the solution program is analyzed.The experimental data in National Aeronautics and Space Administration(NASA)public literature are compared with the simulation results of this exploration,so as to judge the accuracy of the calculation process.The results show that under the same static load,the difference between the minimum film thickness calculated and the test results is not obvious;when the rotor speed of the bearing is 60000 r/min,the influence of the boundary slip effect increases with the increase of the micro groove depth on the flat foil surface;when the eccentricity or the micro groove depth of the bearing increases,the bearing capacity will be strengthened.When the eccentricity is 6µm and 14µm,the viscous friction torque of the new foil bearing increases significantly with the increase of the depth of the foil micro groove,but when the eccentricity is 22µm,the viscous friction torque does not change with the change of the depth of the foil micro groove.It shows that the bearing capacity and performance of foil bearing are improved.

Lagrangian Relaxation Method for Multiobjective Optimization Methods: Solution Approaches

This paper introduces the Lagrangian relaxation method to solve multiobjective optimization problems. It is often required to use the appropriate technique to determine the Lagrangian multipliers in the relaxation method that leads to finding the optimal solution to the problem. Our analysis aims to find a suitable technique to generate Lagrangian multipliers, and later these multipliers are used in the relaxation method to solve Multiobjective optimization problems. We propose a search-based technique to generate Lagrange multipliers. In our paper, we choose a suitable and well-known scalarization method that transforms the original multiobjective into a scalar objective optimization problem. Later, we solve this scalar objective problem using Lagrangian relaxation techniques. We use Brute force techniques to sort optimum solutions. Finally, we analyze the results, and efficient methods are recommended.

Three-dimensional multi-relaxation-time lattice Boltzmann front-tracking method for two-phase flow

We developed a three-dimensional multi-relaxation-time lattice Boltzmann method for incompressible and immiscible two-phase flow by coupling with a front-tracking technique. The flow field was simulated by using an Eulerian grid, an adaptive unstructured triangular Lagrangian grid was applied to track explicitly the motion of the two-fluid interface, and an indicator function was introduced to update accurately the fluid properties. The surface tension was computed directly on a triangular Lagrangian grid, and then the surface tension was distributed to the background Eulerian grid. Three benchmarks of two-phase flow, including the Laplace law for a stationary drop, the oscillation of a three-dimensional ellipsoidal drop, and the drop deformation in a shear flow, were simulated to validate the present model.

Multi-relaxation-time lattice Boltzmann front tracking method for two-phase flow with surface tension

In this paper, an improved incompressible multi-relaxation-time lattice Boltzmann-front tracking approach is proposed to simulate two-phase flow with a sharp interface, where the surface tension is implemented. The lattice Boltzmann method is used to simulate the incompressible flow with a stationary Eulerian grid, an additional moving Lagrangian grid is adopted to track explicitly the motion of the interface, and an indicator function is introduced to update the fluid properties accurately. The interface is represented by using a four-order Lagrange polynomial through fitting a set of discrete marker points, and then the surface tension is directly computed by using the normal vector and curvature of the interface. Two benchmark problems, including Laplace＇s law for a stationary bubble and the dispersion relation of the capillary wave between two fluids are conducted for validation. Excellent agreement is obtained between the numerical simulations and the theoretical results in the two cases.

A CLASS OF GENERALIZED MULTISPLITTING RELAXATION METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

Abstract In this paper,a class of generalized parallel matrix multisplitting relaxation methods for solving linear complementarity problems on the high speed multiprocessor systems is set up.This class of methods not only includes all the existing relaxation methods for the linear complementarity problems,but also yields a lot of novel ones in the sense of multisplitting.We establish the convergence theories of this class of generalized parallel multisplitting relaxation methods under the condition that the system matrix is an H matrix with positive diagonal elements.

分数阶Relaxation-Oscillation方程的一种分数阶预估-校正方法

涉及松弛(Relaxation)和震动(Oscillation)基本现象的过程是与物理密切相关;从数学观点来看,众所周知由时间分数阶导数α,0<α≤1或1<α≤2来控制的现象,被称之为分数阶松弛或分数阶震动现象.本文考虑分数阶Relaxation-Oscillation方程.证明了分数阶Relaxation-Oscillation方程解的存在惟一性,并利用格林函数给出了它的解析解.我们提出一种计算有效的分数阶预估-校正方法,导出了其误差估计.最后给出数值例子.

ASYNCHRONOUS RELAXED ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS OF EQUATIONS

In this paper, the asynchronous versions of classical iterative methods for solving linear systems of equations are considered. Sufficient conditions for convergence of asynchronous relaxed processes are given for H-matrix by which nor only the requirements of [3] on coefficient matrix are lowered, but also a larger region of convergence than that in [3] is obtained.

Multi-relaxation time lattice Boltzmann simulation of inertial secondary flow in a curved microchannel

The inertial secondary flow is particularly important tbr hydrodynamic lbcusing and particle manipulation m biomed- ical research. In this paper, the development of the inertial secondary flow structure in a curved microchannel was investi- gated by the multi relaxation time lattice Boltzmann equation model with a force term. The numerical results indicate that the viscous and inertial competition dominates the development of secondary flow structure development. The Reynolds number, Dean number, and the cross section aspect ratio influence significantly on the development of the secondary vor- texes. Both the intensity of secondary flow and the distance between the normalized vortex centers are functions of Dean numbers but independent of channel curvature radius. In addition, the competition mechanism between the viscous and inertial effects were discussed by performing the particle focusing experiments. The present investigation provides an improved understanding of the development of inertial secondary flows in curved microchannels.

Low-field NMR inversion based on low-rank and sparsity restraint of relaxation spectra

In this paper,we proposed a novel method for low-field nuclear magnetic resonance(NMR)inversion based on low-rank and sparsity restraint(LRSR)of relaxation spectra,with which high quality construction is made possible for one-and two-dimensional low-field and low signal to noise ratio NMR data.In this method,the low-rank and sparsity restraints are introduced into the objective function instead of the smoothing term.The low-rank features in relaxation spectra are extracted to ensure the local characteristics and morphology of spectra.The sparsity and residual term are contributed to the resolution and precision of spectra,with the elimination of the redundant relaxation components.Optimization process of the objective function is designed with alternating direction method of multiples,in which the objective function is decomposed into three subproblems to be independently solved.The optimum solution can be obtained by alternating iteration and updating process.At first,numerical simulations are conducted on synthetic echo data with different signal-to-noise ratios,to optimize the desirable regularization parameters and verify the feasibility and effectiveness of proposed method.Then,NMR experiments on solutions and artificial sandstone samples are conducted and analyzed,which validates the robustness and reliability of the proposed method.The results from simulations and experiments have demonstrated that the suggested method has unique advantages for improving the resolution of relaxation spectra and enhancing the ability of fluid quantitative identification.

ANALYSIS OF WAVEGUIDE PROBLEMS USING A RELAXED ITERATIVE DOMAIN DECOMPOSITION METHOD COMBINED WITH MULTIFRONTAL ALGORITHM

In this paper, an absorbing Fictitious Boundary Condition (FBC) is presented to generate an iterative Domain Decomposition Method (DDM) for analyzing waveguide problems.The relaxed algorithm is introduced to improve the iterative convergence. And the matrix equations are solved using the multifrontal algorithm. The resulting CPU time is greatly reduced.Finally, a number of numerical examples are given to illustrate its accuracy and efficiency.

On the validity of the Boltzmann–BGK model through relaxation evaluation

The Boltzmann-Bhatnagar-Gross-Krook（BGK）model is investigated for its validity regarding the collision term approximation through relaxation evaluation. The evaluation is based on theoretical analysis and numerical comparison between the BGK and direct simulation Monte Carlo（DSMC） results for three specifically designed relaxation problems. In these problems, one or half component of the velocity distribution is characterized by another Maxwellian distribution with a different temperature. It is analyzed that the relaxation time in the BGK model is unequal to the molecular mean collision time. Relaxation of component distribution fails to involve enough contribution from other component distributions, which makes the BGK model unable to capture details of velocity distribution, especially when discontinuity exists in distribution. The BGK model,however, predicts satisfactory results including fluxes during relaxation when the temperature difference is small. Particularly, the model-induced error in the BGK model increases with the temperature difference, thus the model is more reliable for low-speed rarefied flows than for hypersonic flows.