偏微分方程理论和计算暨数学与力学交叉论坛

时间：2024.02.25--03.02

召集人：李海刚、李步扬

会议日程

 报告人题目和摘要

题目：Numerical approximation of discontinuous solutions of the semilinear wave equation

曹家川 香港理工大学

摘要：A fully discrete low-regularity integrator with high-frequency recovery techniques is constructed to approximate rough and possibly discontinuous solutions of the semilinear wave equation. The proposed method can capture the discontinuities of the solutions correctly without spurious oscillations and can approximate rough and discontinuous solutions with a higher convergence rate than pre-existing methods. Rigorous analysis is presented for the convergence rates of the proposed method in approximating solutions such that $(u,\partial_{t}u)\in C([0,T];H^{\gamma}\times H^{\gamma-1})$ for $\gamma\in(0,1]$. For discontinuous solutions of bounded variation in one dimension (which allow jump discontinuities), the proposed method is proved to have almost first-order convergence under the step size condition $\tau \sim N^{-1}$, where $\tau$ and $N$ denote the time step size and the number of Fourier terms in the space discretization, respectively. Extensive numerical examples are presented in both one and two dimensions to illustrate the advantages of the proposed method in improving the accuracy in approximating rough and discontinuous solutions of the semilinear wave equation. The numerical results are consistent with the theoretical results and show the efficiency of the proposed method. This is a joint work with Prof. Li Buyang, Prof. Lin Yanping, and Dr. Yao Fangyan.

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题目：二维弹性力学中的均匀应力夹杂问题

戴明 南京航空航天大学

摘要：报告首先简要介绍了有限变形固体力学的基本框架，而后回顾了二维变形下线弹性力学基本解的复变理论，最后重点讨论了线弹性微观力学中均匀应力夹杂问题的部分研究进展，包括：多个均匀应力夹杂形状的构建方法、多均匀应力夹杂问题中的“静水应力”假设、材料边界对均匀应力夹杂形状的影响等。

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 丁陈森   北京大学  （TBA）

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题目：液晶弹性体的本构、变形和失稳

冯帆    北京大学工学院力学与工程科学系

摘要：液晶弹性体(LCE)是一种对外界光照、温度刺激可以发生长度变化响应的软材料。在本次报告中，我将从液晶弹性体的微观本构出发，讨论其宏观变形，特别是液晶弹性体有别于传统软材料的软弹性(soft elasticity)现象。具体内容包括LCE的力学研究中的两大方面：一是由于液晶排布带来的形状和力学性质的可编程性，如LCE曲线折纸结构的设计；二是由于液晶排布的微结构导致原始非凸本构的均匀化(convexification)带来的有别于传统软材料的新的力学现象，如表面不稳定性等。

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题目：Quantitative homogenization of elliptic problems in periodic high contrast environments

荆文甲  清华大学

摘要：We consider elliptic equations with periodic high contrast coefficients and study the asymptotic analysis when the periodicity is sent to zero and/or the contrast parameters are sent to extreme values. Those coefficients model small inclusions that have very different physical properties compared to the surrounding environment. Homogenization captures the macroscopic effects of those inclusions. We report some quantitative results such as the convergence rates of the homogenization (with proper correctors) and uniform regularity for the solutions of the heterogeneous equations.  The talk is based on joint works with Mr. Xin Fu.

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题目：An efficient multiscale preconditioner for large-scale highly heterogeneous flow

付书彬 宁波东方理工

 摘要：I will introduce an efficient and robust multiscale preconditioner for large-scale Darcy flow in highly heterogeneous porous media. We start from the discretization of the first-order form for the Darcy flow and apply a velocity elimination strategy to obtain an equation with pressure as the only unknown. Then an efficient preconditioner is designed to solve this equation. The key component of the preconditioner is adoption of a non-standard coarse space that contains the media’s heterogeneity information. We solve a carefully constructed spectral problem in each coarse element to form the non-standard coarse space. The experimental results show that our generalized multiscale space based preconditioner is robust with respect to the contrast, size and geometry of the permeability fields.

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题目：On an inverse problem for elastic wave equation.

高忆先    东北师范大学

摘要：This talk focuses on an inverse problem associated with the elastic wave equation . We establish the unique identifying results in simultaneously determining both the unknown density and the internal sources for the plate wave equation  from the passive boundary measurement.  For the  elastic density function, we establish the local Lipschitz stability for low frequencies and the logarithmic stability estimate for high frequencies, respectively.

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题目：Parametric finite element approximations of surface evolution

李步扬  香港理工大学

摘要：We briefly review the origin and development of parametric finite element methods for surface evolution in geometric flow, and the development of artificial tangential motions to improve the mesh quality of the numerically computed surfaces. Then we report some of our recently developed algorithms to improve the mesh quality and our convergence analysis for the algorithms approximating surface evolution. 

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题目：An hp-Adaptive Sampling Algorithm on Dispersion Relation Reconstruction for 2D Photonic Crystals

李光莲  香港大学

摘要：Computing the dispersion relation for two-dimensional photonic crystals is a notoriously challenging task: It involves solving parameterized Helmholtz eigenvalue problems with high-contrast coefficients. To resolve the challenge, we propose a novel hp-adaptive sampling scheme that can detect singular points via adaptive mesh refinement in the parameter domain, and meanwhile, allow for adaptively enriching the local polynomial spaces on the elements that do not contain singular points. In this way, we obtain an element-wise interpolation on an adaptive mesh. We derive an exponential convergence rate when the number of singular points is finite, and a first-order convergence rate otherwise. Numerical tests are provided to illustrate its performance.This is a joint work with my current PhD student Yueqi Wang.

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题目：纤维增强复合材料应力集中的数学研究

李海刚 北京师范大学

摘要：介绍一下高对比度纤维增强复合材料应力集中问题对应的偏微分方程解的梯度估计的近期研究进展和应用。针对周期结构材料，我们建立局部应力集中与整体有效性能之间的统一关系，该结果不仅适用于圆形夹杂、椭圆形夹杂，也适用于一般形状的凸夹杂。

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题目：基于机器学习的多尺度建模：以柔性机械超材料为例

毛晟    北京大学

摘要：固体力学的核心问题之一是建立所研究材料的本构模型用以描述材料的力学响应，而针对具有非均质细观结构的复合材料，其力学响应通常无法用基于均匀性假设的本构模型来表示。因此，这类材料的力学响应往往需要通过多层级、多尺度计算的方式得到，但是当材料的细观结构具有相当的复杂性以及非线性时，这类方法会导致十分巨大的计算量，极大地限制了其在工程实际中的应用。为此，我们结合机器学习的方法，针对这个难点问题进行了一些初步的尝试。我们以柔性机械超材料为主要研究对象，利用深度神经网络和有限元计算，这类材料在静态加载下的宏观等效本构模型，该模型可以很好的预测超材料的分叉及失稳现象。在此基础之上，我们利用图网络，研究了这类材料的宏观动力学行为，并可以很方便地考虑缺陷对该类材料力学行为的影响。在以上的两类问题中，基于机器学习的多尺度方法相比于传统的多尺度方法，其计算效率都有显著提升。

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题目： Recent progress for numerical homogenization

明平兵 中国科学院数学与系统科学研究院

摘要：We shall discuss two topics in the numerical homogenization. 1) Heterogeneous multiscale method (HMM) for general medias that includes the stratified media and the quasi-periodic media, an online-offline method shall be discussed for the localized defects, particularly for the multiple defects and the dynamical problems; 2)Representative volume element for the higher order elliptic problems, particularly for the strain gradient elasticity model for composites.

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题目：Recent results on quantitative homogenization of multi-scale elliptic and parabolic systems

钮维生 安徽大学

摘要： We report some recent results on quantitative homogenization theory, including the convergence rates and the uniform regularity estimates, for the second order elliptic and parabolic systems with multiple scales. The talk is based on joint works with Zhongwei Shen, Yao Xu and Jinping Zhuge.

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题目：材料微观机制的时间多尺度分析

燕鑫 北京航空航天大学 机械工程及自动化学院 

摘要：传统的分子动力学可以描述一系列材料微观的物理、化学、力学机制，为我们认识微观世界打开一扇窗口。然而由于现阶段计算条件的局限性，传统分子动力学最长可以模拟的时间尺度在几十个纳秒范围内。对于一些缓慢的变化过程，比如扩散、缺陷演化等，传统分子动力学在模拟过程中需要人为施加不真实的高温、高压、高应变率等极端载荷。这使得在研究一些对温度、应变率较为敏感的物理过程时，传统分子动力学难以揭示真实的变化机理。本报告将探讨一种从原子精度描述长时间物理现象的新尝试，并结合金属玻璃应力松弛、纳米柱压缩、无定形锂化硅塑形形变等几个实例展示该方法在一些问题中的运用。

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题目：Convergence of an evolving finite element method for surface evolution with tangential motion by harmonic map heat flow

唐榕 香港理工大学

摘要：The finite element approximation of surface evolution under external velocity field is studied. A tangential motion is designed, by using harmonic map heat flow from a reference surface onto the evolving surface, to improve the mesh quality of the numerically computed surface. The convergence of evolving finite element approximations to the surface evolution driven by a specified vector field with the proposed tangential motion is proved for finite elements of degree $k\ge 3$. Numerical examples are provided to demonstrate the convergence of the algorithm and its effectiveness for improving the mesh quality of the numerically computed evolving surface.

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题目：新型纳米材料中离子和电子的输运机制研究和数值模拟

涂斌、卢本卓   国家纳米科学中心、中国科学院数学与系统科学研究院

摘要：本次报告主要介绍两部分内容：（1）简单介绍研究纳米材料中离子和电子输运机制的理论模型以及相关的数值模拟方法，并介绍几个真实纳米材料中的数值模拟实例，例如COF膜中的离子耦合输运，以及机械与电耦合输运，新型纳米半导体器件中的电子与离子的耦合输运；（2）简单介绍机器学习方法，以及如何采用机器学习求解数学物理模型。

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题目：一维结构的波动和振动

王立峰 南京航空航天大学航空航天结构力学及控制全国重点实验室

摘要：报告从弹簧质量系统振动的常微分方程到一维原子链的常微分方程组，再到连续模型的偏微分方程，介绍一维结构的波动和振动问题。进而以典型一维纳米结构和超材料结构为研究对象，针对其波动和振动等问题开展了研究。通过离散结构连续化得到了考虑应变梯度的杆、梁、壳振动的偏微分方程，揭示了碳纳米管的微结构对其波传播的影响。针对考虑应变梯度的非局部弹性结构，发了一种基于移动Kriging插值的无网格方法，用于研究其振动问题。采用多振子结构、超胞结构等设计实现带隙拓宽以应用到低宽频振动抑制中。将被动的超材料结构设计与压电材料以及主动控制方法相结合，利用控制分流电路以及主动控制方法对超材料波动特性进行主动调控，使超材料具有主动可控的能力。最后将介绍一维结构的波动和振动的实验研究工作和研究展望。

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题目：An analytical homogenization framework for dynamic behaviour of heterogeneous materials

王林娟  北京航空航天大学

摘要：The macroscopic governing equations of heterogeneous materials predicted by conventional micromechanical schemes usually have the same form as those of the constituents, that is, if the latter obey the principle of spatial location action, the former also exhibits the same behaviour. While this simple up-scaling transmission of the governing equations is successful in most cases, it cannot describe some fundamental dynamic behaviour of heterogeneous materials, such as the dispersion and bandgaps of elastic waves. In order to solve these problems, we propose a bottom-up dynamic homogenization framework for dynamic mechanical behaviour and transient heat conduction of heterogeneous materials. All parameters of the analytical macroscopic governing equations derived through the homogenization framework are analytically determined by the material and geometrical parameters of the constituents. In addition, the analytical macroscopic governing equations can degenerate into a series of nonlocal thermomechanical models. Furthermore, the fundamental solutions of the analytical spatiotemporal nonlocal macroscopic governing equations are derived, which have two different Green’s functions for initial conditions and body forces. This “double Green’s function phenomena” is first reported, which is due to the spatiotemporal nonlocality of the inertia terms. The analytical solutions of our models agree well with those of direct numerical simulations of dynamic responses of heterogeneous materials. The framework provides an alternative to computational homogenization of heterogeneous materials with much less cost of computation.

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题目：Model reduction methods by the Onsager variational principle

许现民  中国科学院数学与系统科学研究院

摘要：The Onsager variational principle is a fundamental law for irreversible processes in statistic physics. It can be used to develop PDE models for many complicated problems in soft matter. Recently, the principle has been used as an approximation tool to derive reduced models for many multi-scale multi-physics systems. In the first part of the talk, I will present the main idea of the model reduction method by the Onsager principle and the applications in two-phase flows. We derive coarse-grained boundary conditions for moving contact line problems, both with and without contact angle hysteresis. These boundary conditions serve as the foundation for developing efficient numerical methods to solve macroscopic two-phase flow problems. In the second part of the talk, I will present some applications of the method in numerical analysis, including derivation of a diffusion-generated motion method for wetting problems, and moving finite element methods for mean curvature flows and porous media equations. The connection of the method with gradient flows in metric spaces is also addressed.

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题目：力学研究中椭球区域的一些特性及其推广构型

袁天宇    成都大学高等研究院

摘要：从场论的角度来看，大多数物理定律都可以被描述为微分方程，譬如牛顿的万有引力理论、麦克斯韦的电磁场理论、爱因斯坦的广义相对论等等。这些经典理论的预测依赖于相应微分方程边值问题的解，方程的求解难度由微分方程的结构及边界的几何形状决定。椭球构型的边界常常能为求解提供极大的便利性，由此衍生出许多意义深远的结果，以上述三个经典理论为例，均匀椭球区域的牛顿位势在其内部为空间坐标的二次函数；均匀极\磁化的椭球区域内部会产生均匀的电\磁场；爱因斯坦引力场方程在球域外存在封闭形式的施瓦氏解。在近代力学研究中，椭球构型同样展现出特殊的性质，例如，在细观力学夹杂问题中，椭球形状的夹杂就展现出Eshelby均匀特性，由此衍生出一系列创新概念和方法，开启了非均质力学理论高速发展的新时代。鉴于椭球区域在近代力学研究中的关键作用，本工作将从位势理论的角度出发，回顾椭球区域在特定力学问题中的独特性，并在此基础上，介绍椭球区域的一类重要推广—多项式类区域。该类区域产生的力学、物理场能解析、简洁地表示为空间坐标的多项式形式，将为复合材料性能设计和预测提供新的维度。

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题目：Second order Finite element method for 2D Navier-Stokes equations with $L^2$ initial value

张慧   香港理工大学

摘要：We study the error estimate of a fully discrete linearly extrapolated Runge-Kutta method, with finite element method in space, for the two-dimensional Navier-Stokes equations with $L^2$ initial data in convex polygonal domains. By using the discrete semigroup technique and some new estimates, the second-order convergence in time and space is proved for the method. The error estimate is of optimal order for the spatial discretization considered in this article, and currently, the best convergence result obtained for the fully discrete schemes is only first-order in time and first-order in space, which is not optimal in space in the traditional sense.

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题目：流体和弹性体方程的若干变分不等式问题及其数值方法

周冠宇 电子科技大学

摘要：变分不等式可用来描述弹性体的接触问题和自由边界问题，其数值方法是计算力学的一个重要分支。本报告将介绍变分不等式的经典的应用背景、基础理论和一些新的研究进展。对于Navier-Stokes方程的流出边界，我们引入了Signorini型人工边界条件以保证能量不等式成立，由此导出了流体方程的第一类变分不等式问题。我们也考虑流体通过均匀致密孔隙的摩擦型渗透模型，该界面问题是一个第二类变分不等式问题。关于弹性体的裂缝问题，人们提出了关于位移和速度的两类接触界面条件，但是都有数学或物理上的局限性。为此我们考察了一种混合了位移和速度的新型的接触界面条件。针对以上三个偏微分方程的变分不等式问题，我们将详细介绍它们的适定性理论和数值方法。