arXiv subjects

Qiaolin He

Publications and source records attributed to Qiaolin He.

34 records · Page 2Linked to original sources

arXiv2025

The Cauchy Problem for Non-Isentropic compressible Navier-Stokes/Allen-Cahn system with Degenerate Heat-Conductivity

The Cauchy problem for non-isentropic compressible Navier-Stokes/Allen-Cahn system with degenerate heat-conductivity $κ(θ)=\tildeκθ^β$ in 1-d is discussed in this paper. This system is widely used to describe the motion of immiscible two-phase flow in numerical simulation. The wellposedness for strong solution of this problem is established with the $H^1$ initial data for density, temperature, velocity, and the $H^2$ initial data for phase field. The result shows that no discontinuity of the phase field, vacuum, shock wave, mass or heat concentration will be developed at any finite time in the whole space. From the hydrodynamic point of view, this means that no matter how complex the interaction between the hydrodynamic and phase-field effects, phase separation will not occur, but the phase transition is possible.

math.AP↗

arXiv2024

Runge-Kutta Random Feature Method for Solving Multiphase Flow Problems of Cells

Cell collective migration plays a crucial role in a variety of physiological processes. In this work, we propose the Runge-Kutta random feature method to solve the nonlinear and strongly coupled multiphase flow problems of cells, in which the random feature method in space and the explicit Runge-Kutta method in time are utilized. Experiments indicate that this algorithm can effectively deal with time-dependent partial differential equations with strong nonlinearity, and achieve high accuracy both in space and time. Moreover, in order to improve computational efficiency and save computational resources, we choose to implement parallelization and non-automatic differentiation strategies in our simulations. We also provide error estimates for the Runge-Kutta random feature method, and a series of numerical experiments are shown to validate our method.

math.NA↗

arXiv2024

Solving Multi-Group Neutron Diffusion Eigenvalue Problem with Decoupling Residual Loss Function

In the midst of the neural network's success in solving partial differential equations, tackling eigenvalue problems using neural networks remains a challenging task. However, the Physics Constrained-General Inverse Power Method Neural Network (PC-GIPMNN) approach was proposed and successfully applied to solve the single-group critical problems in reactor physics. This paper aims to solve critical problems in multi-group scenarios and in more complex geometries. Hence, inspired by the merits of traditional source iterative method, which can overcome the ill-condition of the right side of the equations effectively and solve the multi-group problem effectively, we propose two residual loss function called Decoupling Residual loss function and Direct Iterative loss function. Our loss function can deal with multi-group eigenvalue problem, and also single-group eigenvalue problem. Using the new residual loss functions, our study solves one-dimensional, two-dimensional, and three-dimensional multi-group problems in nuclear reactor physics without prior data. In numerical experiments, our approach demonstrates superior generalization capabilities compared to previous work.

math.NA↗

arXiv2024

Large Time Behavior of Solutions to Cauchy Problem for 1-D Compressible Isentropic Navier-Stokes/Allen-Cahn System

This paper is concerned with the large time behavior of the solutions to the Cauchy problem for the one-dimensional compressible Navier-Stokes/Allen-Cahn system with the immiscible two-phase flow initially located near the phase separation state. Under the assumptions that the initial data is a small perturbation of the constant state, we prove the global existence and uniqueness of the solutions and establish the time decay rates of the solution as well as its higher-order spatial derivatives. Moreover, we derive that the solutions of the system are time asymptotically approximated by the solutions of the modified parabolic system and obtain decay rates in $L^2$ and $L^1$. Furthermore, we show that the solution of the system is time asymptotically approximated in $L^p (1 \leq p \leq+\infty)$ by the diffusion waves.

math.AP↗

arXiv2024

Deep FBSDE Neural Networks for Solving Incompressible Navier-Stokes Equation and Cahn-Hilliard Equation

Efficient algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the curse of dimensionality. We extend the forward-backward stochastic neural networks (FBSNNs) which depends on forward-backward stochastic differential equation (FBSDE) to solve incompressible Navier-Stokes equation. For Cahn-Hilliard equation, we derive a modified Cahn-Hilliard equation from a widely used stabilized scheme for original Cahn-Hilliard equation. This equation can be written as a continuous parabolic system, where FBSDE can be applied and the unknown solution is approximated by neural network. Also our method is successfully developed to Cahn-Hilliard-Navier-Stokes (CHNS) equation. The accuracy and stability of our methods are shown in many numerical experiments, specially in high dimension.

math.NA↗

arXiv2024

Moving Sampling Physics-informed Neural Networks induced by Moving Mesh PDE

In this work, we propose an end-to-end adaptive sampling neural network (MMPDE-Net) based on the moving mesh method, which can adaptively generate new sampling points by solving the moving mesh PDE. This model focuses on improving the quality of sampling points generation. Moreover, we develop an iterative algorithm based on MMPDE-Net, which makes the sampling points more precise and controllable. Since MMPDE-Net is a framework independent of the deep learning solver, we combine it with physics-informed neural networks (PINN) to propose moving sampling PINN (MS-PINN) and demonstrate its effectiveness by error analysis under some assumptions. Finally, we demonstrate the performance improvement of MS-PINN compared to PINN through numerical experiments of four typical examples, which numerically verify the effectiveness of our method.

math.NA↗

arXiv2024

Large Time Behavior and Sharp Interface Limit of Compressible Navier-Stokes/Allen-Cahn System for Interacting Shock Waves

In this paper, we study the large time behavior and sharp interface limit of the Cauchy problem for compressible Navier-Stokes/Allen-Cahn system with interaction shock waves in the same family. This system is an important mathematical model for describing the motion of immiscible two-phase flow. The results show that, if the initial density and velocity are near the superposition of two shock waves in the same family, then there exists a unique global solution to the compressible Navier-Stokes/Allen-Cahn system, and this solution asymptotically converges to the superposition of the viscous shock wave and rarefaction wave which moving in opposite directions. Moreover, this global-in-time solution converges to the entropy solution of $p$-system in $L^\infty$-norm as the thickness of the diffusion interface tends to zero.

math.AP↗

arXiv2023

Neural Networks Based on Power Method and Inverse Power Method for Solving Linear Eigenvalue Problems

In this article, we propose two kinds of neural networks inspired by power method and inverse power method to solve linear eigenvalue problems. These neural networks share similar ideas with traditional methods, in which the differential operator is realized by automatic differentiation. The eigenfunction of the eigenvalue problem is learned by the neural network and the iterative algorithms are implemented by optimizing the specially defined loss function. The largest positive eigenvalue, smallest eigenvalue and interior eigenvalues with the given prior knowledge can be solved efficiently. We examine the applicability and accuracy of our methods in the numerical experiments in one dimension, two dimensions and higher dimensions. Numerical results show that accurate eigenvalue and eigenfunction approximations can be obtained by our methods.

math.NA↗

arXiv2023

On the uncertainty analysis of the data-enabled physics-informed neural network for solving neutron diffusion eigenvalue problem

In practical engineering experiments, the data obtained through detectors are inevitably noisy. For the already proposed data-enabled physics-informed neural network (DEPINN) \citep{DEPINN}, we investigate the performance of DEPINN in calculating the neutron diffusion eigenvalue problem from several perspectives when the prior data contain different scales of noise. Further, in order to reduce the effect of noise and improve the utilization of the noisy prior data, we propose innovative interval loss functions and give some rigorous mathematical proofs. The robustness of DEPINN is examined on two typical benchmark problems through a large number of numerical results, and the effectiveness of the proposed interval loss function is demonstrated by comparison. This paper confirms the feasibility of the improved DEPINN for practical engineering applications in nuclear reactor physics.

cs.LG↗

arXiv2023

Dynamics of flexible fibers in confined shear flows at finite Reynolds numbers

We carry out a numerical study on the dynamics of a single non-Brownian flexible fiber in two-dimensional Couette flows at finite Reynolds numbers. We employ the bead-spring model of flexible fibers to extend the fluid particle dynamics (FPD) method that is originally developed for rigid particles in viscous liquids. We implement the extended FPD method using a multiple-relaxation-time (MRT) scheme of the lattice Boltzmann method (LBM). The numerical scheme is validated firstly by a series of benchmark simulations that involve liquid-solid coupling. The method is then used to study the dynamics of flexible fibers in Couette flows. We only consider the highly symmetric case where the fibers are placed on the symmetry center of Couette flows and we focus on the effects of the fiber stiffness, the confinement strength, and the finite Reynolds number (from 1 to 10). A diagram of the fiber shape is obtained. For fibers under weak confinement and a small Reynolds number, three distinct tumbling orbits have been identified. (1) Jeffery orbits of rigid fibers. The fibers behave like rigid rods and tumble periodically without any visible deformation. (2) S-turn orbits of slightly flexible fibers. The fiber is bent to an S-shape and is straightened again when it orients to an angle of around 45 degrees relative to the positive x direction. (3) S-coiled orbits of fairly flexible fibers. The fiber is folded to an S-shape and tumbles periodically and steadily without being straightened anymore during its rotation. Moreover, the fiber tumbling is found to be hindered by increasing either the Reynolds number or the confinement strength, or both.

cond-mat.soft↗

arXiv2022

A data-enabled physics-informed neural network with comprehensive numerical study on solving neutron diffusion eigenvalue problems

We present a data-enabled physics-informed neural network (DEPINN) with comprehensive numerical study for solving industrial scale neutron diffusion eigenvalue problems (NDEPs). In order to achieve an engineering acceptable accuracy for complex engineering problems, a very small amount of prior data from physical experiments are suggested to be used, to improve the accuracy and efficiency of training. We design an adaptive optimization procedure with Adam and LBFGS to accelerate the convergence in the training stage. We discuss the effect of different physical parameters, sampling techniques, loss function allocation and the generalization performance of the proposed DEPINN model for solving complex problem. The feasibility of proposed DEPINN model is tested on three typical benchmark problems, from simple geometry to complex geometry, and from mono-energetic equation to two-group equations. Numerous numerical results show that DEPINN can efficiently solve NDEPs with an appropriate optimization procedure. The proposed DEPINN can be generalized for other input parameter settings once its structure been trained. This work confirms the possibility of DEPINN for practical engineering applications in nuclear reactor physics.

physics.comp-ph↗

arXiv2022

Sharp Interface Limit for Compressible Immiscible Two-Phase Dynamics with Relaxation

In this paper, the compressible immiscible two-phase flow with relaxation is investigated, this model can be regarded as a natural modification of Jin-Xin relaxation scheme proposed and developed by S.Jin and Z.P.Xin([Comm.Pure Appl.Math., 48,1995]) in view of the numerical approximation of conservation laws. Given any entropy solution consists of two different families of shocks interacting at some positive time for the standard two-phase compressible Euler equations, it is proved that such entropy solution is the sharp interface limit for a family global strong solutions of the modified Jin-Xin relaxation scheme for Navier-Stokes/Allen-Cahn system, here the relaxation time is selected as the thickness of the interface, weighted estimation and improved antiderivative method are used in the proof. Moreover, the simulation results are given by this modified Jin-Xin relaxation scheme method. Both numerical and theoretical results show that, the interacting shock waves can pass through the interface without any effect.

math.AP↗

arXiv2020

The Fictitious Domain Method Based on Navier Slip Boundary Condition for Simulation of Flow-Particle Interaction

In this article, we develop a least--squares/fictitious domain method for direct simulation of fluid particle motion with Navier slip boundary condition at the fluid--particle interface. Let $Ω$ and $B$ be two bounded domains of $\mathbb{R}^{d}$ such that $\overline{B} \subset Ω$. The motion of solid particle $B$ is governed by Newton's equations. Our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the full $Ω$, followed by a well--chosen correction over $B$ and corrections related to translation velocity and angular velocity of the particle. This method is of the virtual control type and relies on a least--squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Since the fully explicit scheme to update the particle motion using Newton's equation is unstable, we propose and implement an explicit--implicit scheme in which, at each time step, the position of the particle is updated explicitly, and the solution of Navier-Stokes equations and particle velocities are solved by the the least--squares/fictitious domain method implicitly. Numerical results are given to verify our numerical method.

math.NA↗

arXiv2020

Global Strong Solution to a Thermodynamic Compressible Diffuse Interface Model with Temperature Dependent Heat-conductivity in 1-D

In this paper, we investigate the wellposedness of the non-isentropic compressible Navier-Stokes/Allen-Cahn system with the heat-conductivity proportional to a positive power of the temperature. This system describes the flow of a two-phase immiscible heat-conducting viscous compressible mixture. The phases are allowed to shrink or grow due to changes of density in the fluid and incorporates their transport with the current. We established the global existence and uniqueness of strong solutions for this system in 1-D, which means no phase separation, vacuum, shock wave, mass or heat or phase concentration will be developed in finite time, although the motion of the two-phase immiscible flow has large oscillations and the interaction between the hydrodynamic and phase-field effects is complex. Our result can be regarded as a natural generalization of the Kazhikhov-Shelukhin's result ([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)]) for the compressible single-phase flow with constant heat conductivity to the non-isentropic compressible immiscible two-phase flow with degenerate and nonlinear heat conductivity.

math.AP↗

arXiv2018

Global Solution for Gas-Liquid Flow of 1-D van der Waals Equation of State with Large Initial Data

This paper is concerned with a diffuse interface model for the gas-liquid phase transition. The model consists the compressible Navier-Stokes equations with van der Waals equation of state and a modified Allen-Cahn equation. The global existence and uniqueness of strong solution with the periodic boundary condition (or the mixed boundary condition) in one dimensional space is proved for large initial data. Furthermore, the phase variable and the density of the gas-liquid mixture are proved to stay in the physical reasonable interval. The proofs are based on the elementary energy method and the maximum principle, but with new development, where some techniques are introduced to establish the uniform bounds of the density and to treat the non-convexity of the pressure function.

math.AP↗

arXiv2018

Asymptotic Stability of Solutions for 1-D Compressible Navier-Stokes-Cahn-Hilliard system

This paper is concerned with the evolution of the periodic boundary value problem and the mixed boundary value problem for a compressible mixture of binary fluids modeled by the Navier-Stokes-Cahn-Hilliard system in one dimensional space. The global existence and the large time behavior of the strong solutions for these two systems are studied. The solutions are proved to be asymptotically stable even for the large initial disturbance of the density and the large velocity data. We show that the average concentration difference for the two components of the initial state determines the long time behavior of the diffusive interface for the two-phase flow.

math.AP↗