Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Stress tensor

A full implementation of the analytical stress tensor for periodic systems is reported in the TURBOMOLE program package within the framework of Kohn–Sham density functional theory using Gaussian-type orbitals as basis functions. It is the extension of the implementation of analytical energy gradients (Lazarski et al., Journal of Computational Chemistry 2016, 37, 2518–2526) to the stress tensor for the purpose of optimization of lattice vectors. Its key component is the efficient calculation of the Coulomb contribution by combining density fitting approximation and continuous fast multipole method. For the exchange-correlation (XC) part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chemical Theory and Computation 2011, 7, 3097–3104) is extended to XC weight derivatives and stress tensor. The computational efficiency and favorable scaling behavior of the stress tensor implementation are demonstrated for various model systems. The overall computational effort for energy gradient and stress tensor for the largest systems investigated is shown to be at most two and a half times the computational effort for the Kohn–Sham matrix formation. © 2019 Wiley Periodicals, Inc.     

Versatile anisotropic mesh adaptation methodology applied to pure quantity of interest error estimator. Steady,laminar incompressible flow

We introduce a new flexible mesh adaptation approach to efficiently compute a quantity of interest by the finite element method. Efficiently, we mean that the method provides an evaluation of that quantity up to a predetermined accuracy at a lower computational cost than other classical methods. The central pillar of the method is our scalar error estimator based on sensitivities of the quantity of interest to the residuals. These sensitivities result from the computation of a continuous adjoint problem. The mesh adaptation strategy can drive anisotropic mesh adaptation from a general scalar error contribution of each element. The full potential of our error estimator is then reached. The proposed method is validated by evaluating the lift, the drag, and the hydraulic losses on a 2D benchmark case: the flow around a cylinder at a Reynolds number of 20.     

On inexact alternating direction implicit iteration for continuous Sylvester equations

In this paper, we study the alternating direction implicit (ADI) iteration for solving the continuous Sylvester equation AX + XB = C , where the coefficient matrices A and B are assumed to be positive semi‐definite matrices (not necessarily Hermitian), and at least one of them to be positive definite. We first analyze the convergence of the ADI iteration for solving such a class of Sylvester equations, then derive an upper bound for the contraction factor of this ADI iteration. To reduce its computational complexity, we further propose an inexact variant of the ADI iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer ADI iteration. The convergence is also analyzed in detail. The numerical experiments are given to illustrate the effectiveness of both ADI and inexact ADI iterations.     

Topological entropy of continuous self-maps on closed surfaces

The objective of this work is to present sufficient conditions for having positive topological entropy for continuous self-maps defined on a closed surface by using the action of this map on the homological groups of the closed surface.     

一种改进的轨道动力学模型

航天器的矢径可以分解为矢径模和单位矢量的乘积,利用该性质将传统轨道动力学方程分解为矢径模和矢径方向的动力学方程组,实现了航天器位置信息的分离;针对两个方程分别采用常数变易法和四元数描述方法,将轨道动力学模型转化为线性无奇异的方程组,同时得到了7 个新轨道变量,且建立了新轨道变量与惯性系下航天器位置速度信息以及轨道六要素之间的相互转换关系. 该轨道模型适用于任意形式的推力和摄动,避免了奇异性,且在虚拟时间的意义下,航天器的旋转角速度只取决于法向力;在常值推力和变推力的情况下,对该模型进行了数值验证,验证了新模型的可适用性、数值稳定性以及计算精度高的优势.     

Hilbert空间中连续广义标架的和

基于Hilbert空间标架理论，借助算子工具，用己有连续广义标架构造了新连续广义标架，并给出了有限个连续广义标架和构成新连续广义标架的充要条件，为构造连续广义标架提供了新方法。     

Runge–Kutta Methods for Systems of Differential Equation with Piecewise Continuous Arguments: Convergence and Stability

This work deals with the convergence and stability of Runge–Kutta methods for systems of differential equation with piecewise continuous arguments x^′(t) = Px(t)+Qx([t+1∕2]) under two cases for coe?cient matrix. First, when P and Q are complex matrices, the su?cient condition under which the analytic solution is asymptotically stable is given. It is proven that the Runge–Kutta methods are convergent with order p. Moreover, the su?cient condition under which the analytical stability region is contained in the numerical stability region is obtained. Second, when P and Q are commutable Hermitian matrices, using the theory of characteristic, the necessary and su?cient conditions under which the analytic solution and the numerical solution are asymptotically stable are presented, respectively. Furthermore, whether the Runge–Kutta methods preserve the stability of analytic solution are investigated by the theory of Padé approximation and order star. To demonstrate the theoretical results, some numerical experiments are adopted.     

Spectral theory of one-channel operators and application to absolutely continuous spectrum for Anderson type models

A one-channel operator is a self-adjoint operator on ${?}^2(\mathbb{G})$ for some countable set $\mathbb{G}$ with a rank 1 transition structure along the sets of a quasi-spherical partition of $\mathbb{G}$. Jacobi operators are a very special case. In essence, there is only one channel through which waves can travel across the shells to infinity. This channel can be described with transfer matrices which include scattering terms within the shells and connections to neighboring shells. Not all of the transfer matrices are defined for some countable set of energies. Still, many theorems from the world of Jacobi operators are translated to this setup. The results are then used to show absolutely continuous spectrum for the Anderson model on certain finite dimensional graphs with a one-channel structure. This result generalizes some previously obtained results on antitrees.     

连续梁在周期荷载下的动力稳定性简化分析方法

对于一般任意支撑的连续梁结构动力稳定性问题，已有的计算方法求解过程都很复杂，给工程设计带来极大的不便．本文提出了一个简化的分析方法，利用现有的商业软件，只需求得连续梁的自然频率及静力屈曲（失稳）荷载，就可容易得到结构的动力失稳区域，当考虑结构阻尼对不稳定区域的影响时，可将阻尼矩阵表达为Rayleigh阻尼的形式．研究结果表明：采用本文计算方法与已有的理论计算方法得到的连续梁主参数共振的不稳定边界非常吻合，而本文计算方法更为简单，计算结果可靠，计算精度高，可满足工程设计的需要．