项目编号：天元地区发展论坛DQ201804
会议时间：2018年7月11日-12日
会议地点：新疆乌鲁木齐雪莲酒店
日程安排：
2018年7月11日 |
10:00-10:20 开幕式及合影 |
Session 1（主席：聂玉峰 教授） |
10:20-11:00 黄艾香 教授 (西安交通大学) Post-Galerkin Methods Based on an Approximate Inertial Manifold |
11:00-11:40李若 教授（北京大学） Finding 13-Moment System Beyond Grad |
Session 2（主席：李若 教授） |
12:00-12:40 郑伟英 教授（中科院数学与系统科学研究院） A charge-conservative mixed finite element method for inductionless and incompressible equations |
12:40-13:20 应文俊 教授 (上海交通大学) Recent Developments of a Potential Theory based Cartesian Grid Method for Partial Differential Equations |
Session 3（主席：郑伟英 教授） |
15:30-16:10李开泰 教授 (西安交通大学) 球单元和球壳单元有限元及其在弹性力学和流体力学中的应用 |
16:10-16:50聂玉峰 教授 （西北工业大学） 泡泡布点方法及其应用研究进展 |
16：50-17：20 郑海标 副教授 （华东师范大学） A new coupled model and stabilized decoupled numerical method for closed-loop geothermal system |
Session 4（主席：何银年 教授） |
17:40-18:20李杰权 教授（北京应用物理与计算数学研究所） High Order Temporal-Spatially Coupled and Thermodynamically Consistent GRP Schemes for Compressible Multi-fluid Flows |
18:20-19:00 张庆海 教授（浙江大学） Interface tracking close to machine precision and the best estimates of interface curvature |
19:00-19:30 黄鹏展 副教授 （新疆大学数学与系统科学学院） An efficient two-level algorithm for the 2D/3D stationary incompressible magnetohydrodynamics |
2018年7月12日 |
Session 5（主席：张庆海 教授） |
09:30-10:10 刘铁刚 教授 （北京航空航天大学） 带加速修正的MGFM方法（MGFM+A） |
10:10-10:50 张辉 教授 （北京师范大学） A positivity-preserving, energy stable and convergent numerical scheme for the Cahn-Hilliard equation with a Flory-Huggins-deGennes energy |
Session 6（主席：张辉 教授） |
11:10-11:50 赵建平 副教授(新疆大学) Macroscopic model for head-on binary droplet collisions in a gaseous medium |
11:50-12:30 邓伟华 教授 （兰州大学） Modelling and Simulation for Anomalous and Non-ergodic Diffusion |
12:30-13:10 何银年 教授 （西安交通大学） Decoupled Finite Element Methods for the 3D Primitive Equations of Ocean |
13:10-13:30 闭幕式 |
15:30-19:00 座谈与讨论 |
报告摘要
Post-Galerkin Methods Based on an Approximate Inertial Manifold
黄艾香教授，西安交通大学
摘要： 1 Introduction
2 Projection
3 Lower Frequency Components
4 Approximate Inertial Manifold and Higher Frequency Components
5 Post-Galerkin Approximation Solution
6 Numerical Example
Finding 13-Moment System Beyond Grad
李若教授，北京大学
摘要：We point out that the thermodynamic equilibrium is not an interior point of the hyperbolicity region of Grad's 13-moment system. With a compact expansion of the phase density, which is compacter than Grad's expansion, we derived a modified 13-moment system. The new 13-moment system admits the thermodynamic equilibrium as an interior point of its hyperbolicity region. We deduce a concise criterion to ensure the hyperbolicity, thus the hyperbolicity region can be quantitatively depicted.
A charge-conservative mixed finite element method for inductionless and incompressible equations
郑伟英研究员，中科院数学与系统科学研究院
摘要：We propose a charge-conservative mixed finite element method for inductionless and incompressible magnetohydrodynamic (MHD) equations. The discrete current density satisfies the divergence-free constraint exactly. A robust and quasi-optimal solver is proposed to solve the discrete problem by the preconditioning of the MHD differential operators. The preconditioning for the stiffness matrix is then obtained by transferring the operator preconditioning to its algebraic counterpart. The discrete solver is optimal in the sense that the number of iterations is uniform with respect to mesh refinements. By extensive numerical examples for both stationary and time-dependent MHD problems, we demonstrate the robustness of the solver to Reynolds number and the optimality of the solver to the number of unknowns.
Recent Developments of a Potential Theory based Cartesian Grid Method for Partial Differential Equations
应文俊教授，上海交通大学
摘要：I will talk on recent developments of a potential theory based Cartesian grid method for partial differential equations. The method is a generalization of the traditional boundary integral method for elliptic PDEs and works for variable coefficients and nonlinear PDEs. It is a Cartesian grid method. It shares general advantages with other Cartesian grid methods. Grid generation is easy and cheap. Discrete equations on Cartesian grids can be efficiently solved with fast algorithms such as an FFT based direct solver or a geometric multigrid iterative solver. It is convenient to work with moving interface and free boundary problems. This talk will present recent development of a fourth-order version of the method for the biharmonic equation as well as its application in computation of incompressible fluids.
球单元和球壳单元有限元及其在弹性力学和流体力学中的应用
李开泰教授，西安交通大学
摘要：给出有六个网格节点的球单元或球壳单元的形状函数，他是由球函数构造的。
应用于：线性和非线性弹性力学的结构强度分析，特别是由3D 打印所生成的复合弹性材料； 应用于三维球的 Stokes 绕流和 两个同心旋转球壳内的三维粘性流动的Navier-Stokes 方程有限元逼近，研究他的Taylor 的基本流和他的分歧问题。
泡泡布点方法及其应用研究进展
聂玉峰 教授, 西北工业大学
摘要：简要介绍泡泡布点方法原理及其在曲面、区域的各向同性、异性网格生成研究进展，及其在并行生成和自适应有限元方法中的应用研究进展。
A new coupled model and stabilized decoupled numerical
method for closed-loop geothermal system
郑海标副教授，华东师范大学
摘要：This talk is to propose a new coupled multi-physics model and a decoupled stabilized finite element method for the closed-loop type geothermal system, which mainly consists of a network of underground heat exchange pipelines to transfer the geothermal heat from the geothermal reservoir. The new mathematical model considers the heat transfer between two different flow regions, namely the porous media flow region in the geothermal reservoir and the free flow region in the pipes. Darcy’s law and Stokes equations are considered to govern the flows in these two regions, respectively, while the heat equation is coupled with the flow equations to describe the heat transfer in both regions. Furthermore, on the interface between the two regions, four physically valid interface conditions are considered to describe the continuity of the temperature and the heat flux as well as the no-fluid communication feature of the closed-loop geothermal system. In the variational formulation, a stabilization term with a stabilization parameter is added to ensure the stability. To solve the proposed model accurately and efficiently, we develop a decoupled stabilized finite element method which decouples not only the two flow regions but also the heat field and the flow field in each region. The stability of the proposed method is proved. Three numerical experiments are provided to validate and illustrate the proposed model and numerical method.
High Order Temporal-Spatially Coupled and
Thermodynamically Consistent GRP Schemes for Compressible Multi-fluid Flows
李杰权研究员，北京应用物理与计算数学研究所
摘要：The most distinct feature of compressible multi-fluid flows is the presence of singularities (shocks, material interfaces, vortices and other discontinuities etc) in flows, which arises notorious difficulties in all aspects of theoretical justification, numerical analysis, scientific computation as well as engineering applications. Just from the viewpoint of the design of numerical methods, high resolution schemes has become the mainstream for several decades, however, there are still many bottleneck problems unsolved. This lecture will address some fundamentals in this aspect, based the celebrated Lax-Wendroff method, which can be traced back traced back to Cauchy-Kowalevski’s theory in 1700's in terms of power series solution for hyperbolic problems. The irreplaceable values the Lax-Wendroff approach can be summarized as follows.
Interface tracking close to machine precision
and the best estimates of interface curvature
张庆海教授，浙江大学
摘要：Most interface tracking IT methods are at best second-order accurate. The most serious disadvantage of explicit IT methods, however, is probably the absence of an analytic framework. Based on the donating region theory, and a topological space that models physically meaningful material regions, we resolve this deficit by proposing a generic framework via Mapping and Adjusting Regular Semialgebraic sets (MARS). Using MARS, we formally proved the second-order accuracy of Volume-of-Fluid methods, clarified many subtle issues such as the accuracy deterioration caused by local C1 discontinuities, and analyzed other explicit IT methods such as moment-of-Fluid methods and front-tracking methods. Also inspired by MARS, we proposed a new IT method which achieved 4th-, 6th-, and 8th-order accuracy for an arbitrary number of phases. For the classic vortex-shear tests, our new method achieves close to machine precision on a 128-by-128 grid. Finally, we showed under the framework of MARS that there exists a best estimation of curvature at any regular point of the interface, which can be easily obtained by our new eighth-order algorithms.
An efficient two-level algorithm for the 2D/3D stationary incompressible magnetohydrodynamics
黄鹏展副教授，新疆大学数学与系统科学学院
摘要：In the talk, we show a two-level finite element algorithm for solving the 2D/3D
stationary incompressible magnetohydrodynamics based on the Newton iterative method.
This algorithm is consisting of solving one nonlinear system on a coarse mesh with mesh
size H and two linearized problems with different loads on a fine mesh with mesh size h.
Compared with existing work on the two-level method for the MHD model, our two-level
method allows a much high order scaling between the coarse and fine grid sizes. Furthermore,stability and convergence of this present method are analyzed. Finally, the applicability and effectiveness of the present algorithm are illustrated by several numerical experiments.
带加速修正的MGFM方法（MGFM+A）
刘铁刚教授，北京航空航天大学
摘要：通过建立广义多介质Riemann问题分析了各种虚拟介质（GFM）类方法在应用到处理带加速度的多介质物质界面时遇到的问题，给出了带加速修正的MGFM方法（MGFM+A），新的方法可以很好地克服目前GFM类方法处理带加速度物质界面所遇到的问题。
A positivity-preserving, energy stable and convergent numerical scheme for the Cahn-Hilliard equation with a Flory-Huggins-deGennes energy
张辉教授，北京师范大学
摘要：This talk is focused on the bound estimate and convergence analysis of an unconditionally energy stable scheme for the MMC-TDGL equation, a Cahn-Hilliard equation with a Flory-Huggins-deGennes energy. The numerical scheme, a finite difference algorithm based on a convex splitting technique of the energy functional, was proposed in [Sci. China Math. 59(2016),1815]. We provide a theoretical justification of the unique solvability for the proposed numerical scheme, in which a well-known difficulty associated with the singular nature of the logarithmic energy potential has to be handled. Meanwhile, a careful analysis reveals that, such a singular nature prevents the numerical solution of the phase variable reaching the limit singular values, so that the positivity preserving property could be proved at a theoretical level. In particular, the natural structure of the deGennes diffusive coefficient also ensures the desired positivity-preserving property. In turn, the unconditional energy stability becomes an outcome of the unique solvability and the convex-concave decomposition for the energy functional. Moreover, an optimal rate convergence analysis is presented.
Macroscopic model for head-on binary droplet collisions in a gaseous medium
赵建平副教授，新疆大学
摘要：In this work,coalescence-bouncing transitions of head-on binary
droplet collisions are predicted by a novel macroscopic model based entirely on fundamental laws of physics. By making use of the lubrication theory of Zhang and Law [Phys. Fluids 23,042102 (2011)], we have modified the Navier-Stokes equations to accurately account for the rarefied nature of the interdroplet gas film. Through the disjoint pressure model, we have incorporated the intermolecular Van der Waals forces. Our model does not use any adjustable (empirical) parameters. It therefore encompasses an extreme range of length scales (more than 5 orders of magnitude): from those of the external flow in excess of the droplet size (a few hundred micros) to the effective range of the Van der Waals force around 10 nm. A state of the art moving adaptive mesh method, capable of resolving all the relevant length scales, has been employed. Our numerical simulations are able to capture the coalescence-bouncing and bouncing -coalescence transitions that are observed as the collision intensity increases. The predicted transition Weber numbers for tetradecane and water droplet collisions at different pressures show remarkably good agreement with published experimental values. Our study also sheds new light on the roles of gas density, droplet size and mean free path in the rupture of the gas film.
Modelling and Simulation for Anomalous and Non-ergodic Diffusion
邓伟华教授，兰州大学
摘要：For the particles undergoing the anomalous diffusion with different waiting time distributions for different internal states, we derive the Fokker-Planck and Feymann-Kac equations, respectively, describing positions of the particles and functional distributions of the trajectories of particles; in particular, the equations governing the functional distribution of internal states are also obtained. The dynamics of the stochastic processes are analyzed and the applications, calculating the distribution of the first passage time and the distribution of the fraction of the occupation time, of the equations are given. For the further application of the newly built models, we make very detailed discussions on the none-immediately-repeated stochastic process, e.g., the random walk of smart animals, and its boundary issues.
Decoupled Finite Element Methods for the 3D Primitive Equations of Ocean
何银年教授，西安交通大学
摘要：In this paper, two decoupled finite element methods are proposed for solving the 3D primitive equations of ocean. Based on the finite element approximation, optimal error estimates are given under the convergence condition. And the detailed algorithms are given in the section of numerical tests. Further, numerical calculations are implemented to validate the theoretical analysis and more calculations are implemented for a more meaningful problem. For both theoretical and numerical points of view, the proposed decoupled finite element methods are the effective strategies to solve the 3D primitive equations of ocean.