基于大涡模拟的移动粒子半隐式法研究

将大涡模拟(LES)和无网格的移动粒子半隐式法(MPS)相结合, 以求解湍流中的自由表面问题. 对N-S方程进行滤波计算可得到大涡模拟的控制方程, 大涡模拟的控制方程相对于以往的移动粒子半隐式法而言仅多出雷诺应力项, 通过亚粒子应力(sub-particle-scale,SPS)模型并引入Smagorinsky涡黏模型将雷诺应力模型化, 可实现移动粒子半隐式法的大涡模拟. 将MPS-LES应用至具有大变形自由表面的共振晃荡中, 其模拟结果同实验及其他数值模拟结果都相当接近.      

A high-order parallel Eulerian-Lagrangian algorithm for advection-diffusion problems on unstructured meshes

In this paper, we present a high-order discontinuous Galerkin Eulerian-Lagrangian method for the solution of advection-diffusion problems on staggered unstructured meshes in two and three space dimensions. The particle trajectories are tracked backward in time by means of a high-order representation of the velocity field and a linear mapping from the physical to a reference system, hence obtaining a very simple and efficient strategy that permits to follow the Lagrangian trajectories throughout the computational domain. The use of an Eulerian-Lagrangian discretization increases the overall computational efficiency of the scheme because it is the only explicit method for the discretization of convective terms that admits large time steps without imposing a Courant-Friedrichs-Lewy–type stability condition. This property is fully exploited in this work by relying on a semi-implicit discretization of the incompressible Navier-Stokes equations, in which the pressure is discretized implicitly; thus, the sound speed does not play any role in the restriction of the maximum admissible time step. The resulting mild Courant-Friedrichs-Lewy stability condition, which is based only on the fluid velocity, is here overcome by the adoption of the Eulerian-Lagrangian method for the advection terms and an implicit scheme for the diffusive part of the governing equations. As a consequence, the novel algorithm is able to run simulation with a time step that is defined by the user, depending on the desired efficiency and time scale of the physical phenomena under consideration. Finally, a complete Message Passing Interface parallelization of the code is presented, showing that our approach can reach up to 96% of scaling efficiency.     

A Second-Order Semi-Implicit Method for the Inertial Landau-Lifshitz-Gilbert Equation

Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization. Its dynamics spans over a wide range of timescales from femtosecond to picosecond, and then to nanosecond. The Landau-Lifshitz-Gilbert (LLG) equation has been widely used in micromagnetics simulations over decades. Recent theoretical and experimental advances have shown that the inertia of magnetization emerges at sub-picosecond timescales and contributes significantly to the ultrafast magnetization dynamics, which cannot be captured intrinsically by the LLG equation. Therefore, as a generalization, the inertial LLG (iLLG) equation is proposed to model the ultrafast magnetization dynamics. Mathematically, the LLG equation is a nonlinear system of parabolic type with (possible) degeneracy. However, the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy, and exhibits more complicated structures. It behaves as a hyperbolic system at sub-picosecond timescales, while behaves as a parabolic system at larger timescales spanning from picosecond to nanosecond. Such hybrid behaviors impose additional difficulties on designing efficient numerical methods for the iLLG equation. In this work, we propose a second-order semi-implicit scheme to solve the iLLG equation. The second-order temporal derivative of magnetization is approximated by the standard centered difference scheme, and the first-order temporal derivative is approximated by the midpoint scheme involving three time steps. The nonlinear terms are treated semi-implicitly using one-sided interpolation with second-order accuracy. At each time step, the unconditionallyunique solvability of the unsymmetric linear system is proved with detailed discussions on the condition number. Numerically, the second-order accuracy of the proposed method in both time and space is verified. At sub-picosecond timescales, the inertial effect of ferromagnetics is observed in micromagnetics simulations, in consistency with the hyperbolic property of the iLLG model; at nanosecond timescales, the results of the iLLG model are in nice agreements with those of the LLG model, in consistency with the parabolic feature of the iLLG model.     

基于MPS-FEM耦合方法对比研究刚性与弹性挡板对液舱晃荡的抑制作用

由流体冲击载荷引起的流固耦合问题广泛存在于船舶与海洋工程领域.例如:在特定激励频率下载液货舱内流体的非线性运动引起对舱壁的砰击作用,进而可能影响液舱围护系统的安全性.由于此类流固耦合问题通常涉及多学科知识,且流体自由面的变化具有强非线性特征,对研究人员带来较大挑战.考虑到Lagrange类方法在处理结构和流体自由面大变形问题上的优势,基于MPS-FEM耦合方法开发了流固耦合求解器.其中,采用MPS方法来数值模拟流体场瞬态变化,FEM方法来分析结构场的变形问题.此外,该求解器采用了弱耦合的方式来实现流体场和结构场之间的数据传递.为了验证该方法在处理流固耦合问题上的可靠性,首先数值研究了溃坝泄洪流与弹性挡板之间的流固耦合标准算例,数值结果与实验标准结果能够较好地吻合.此后,采用该求解器数值研究了带刚性挡板和弹性挡板的液舱晃荡问题,对比分析了多种激励频率下两种挡板对液舱内流体运动及舱壁上冲击压力的抑制效果.     

A Simple Semi-Implicit Scheme for Partial Differential Equations with Obstacle Constraints

We develop a simple and efficient numerical scheme to solve a class of obstacle problems encountered in various applications. Mathematically, obstacle problems are usually formulated using nonlinear partial differential equations (PDE). To construct a computationally efficient scheme, we introduce a time derivative term and convert the PDE into a time-dependent problem. But due to its nonlinearity, the time step is in general chosen to satisfy a very restrictive stability condition. To relax such a time step constraint when solving a time dependent evolution equation, we decompose the nonlinear obstacle constraint in the PDE into a linear part and a nonlinear part and apply the semi-implicit technique. We take the linear part implicitly while treating the nonlinear part explicitly. Our method can be easily applied to solve the fractional obstacle problem and min curvature flow problem. The article will analyze the convergence of our proposed algorithm. Numerical experiments are given to demonstrate the efficiency of our algorithm.     

带有摩擦耗能元件的框架结构动力分析

本文提出粘性屈服模型来模拟摩擦耗能元件的力-速度关系,该模型是连续变化的,克服了库仑摩擦模型不连续导致数值计算复杂的缺点,在进行摩擦耗能体系的动力分析中,采用缩减自由度技术,并作适当的变换,则带有摩擦耗能元件体系的动力分析归结为求解微分代数方程,本文采用增量型Rosenbrock二级三阶半隐式Runge-Kutta法求解该方程,以考虑框架和支撑的材料和几何非线性。对带有摩擦耗能元件的钢框架进行了弹塑性动力分析,研究了支撑刚度与结构层刚度的比值、摩擦力的大小以及地震波类型等参数对体系的影响。     

Development and elaboration of numerical method for simulating gas–liquid–solid three-phase flows based on particle method

A numerical method for simulating gas–liquid–solid three-phase flows based on the moving particle semi-implicit (MPS) approach was developed in this study. Computational instability often occurs in multiphase flow simulations if the deformations of the free surfaces between different phases are large, among other reasons. To avoid this instability, this paper proposes an improved coupling procedure between different phases in which the physical quantities of particles in different phases are calculated independently. We performed numerical tests on two illustrative problems: a dam-break problem and a solid-sphere impingement problem. The former problem is a gas–liquid two-phase problem, and the latter is a gas–liquid–solid three-phase problem. The computational results agree reasonably well with the experimental results. Thus, we confirmed that the proposed MPS method reproduces the interaction between different phases without inducing numerical instability.     

Stability of Semi-implicit Finite Volume Scheme for Level Set Like Equation

We study numerical methods for level set like equations arising in image processing and curve evolution problems. Semi-implicit finite volume-element type schemes are constructed for the general level set like equation(image selective smoothing model) given by Alvarez et al.(Alvarez L, Lions P L, Morel J M. Image selective smoothing and edge detection by nonlinear diffusion II. SIAM J. Numer.Anal., 1992, 29: 845–866). Through the reasonable semi-implicit discretization in time and co-volume method for space approximation, we give finite volume schemes,unconditionally stable in L∞ and W1,2(W1,1) sense in isotropic(anisotropic) diffusion domain.