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Incompatible deformation field and Riemann curvature tensor

Applied Mathematics and Mechanics(English Edition) 2017年第3期38卷 311-332页

作者： Bohua SUN Department of Mechanical Engineering Cape Peninsula University of Technology Cape Town 7535 South Africa

compatibility conditions of a deformation field in continuum mechanics have been revisited via two different routes. One is to use the deformation gradient, and the other is a pure geometric one. Variations of the displacement vector and the displacement density tensor are obtained explicitly in terms of the Riemannian curvature tensor. The explicit relations reconfirm that the compatibility condition is equivalent to the vanishing of the Riemann curvature tensor and reveals the non-Euclidean nature of the space in which the dislocated continuum is imbedded. Comparisons with the theory of Kr¨oner and Le-Stumpf are provided.

关键词： compatibility condition Riemann curvature tensor deformation gradient Burgers vector dislocation density tensor

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Recasting theory of elasticity with micro-finite elements

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Acta Mechanica Sinica 2015年第5期31卷 679-684页

作者： Wennan Zou Institute for Advanced Study Nanchang University

In the classical theory of elasticity,a body is initially modeled as a homogeneous and dense assemblage of constituent ＂material particles＂.The kernel concept of elastic deformation is the displacement of the particle that associates the current configuration with the reference *** this paper,we exploit an alternative constituent ＂micro-finite element＂,and use the stretch of the element as the essential quality to recast the theory of *** should be realized that such a treatment means that the elastic body can be modeled as a finite covering of elements and consequently characterized by a *** recasting of the elasticity theory becomes more feasible for dealing with defects and topological evolution.

关键词： Micro-finite element Stretch Displacement Relative rotation compatibility condition

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A Mathematical Comparison of the Schwarzschild and Kerr Metrics

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Journal of Modern Physics 2020年第10期11卷 1672-1710页

作者： J.-F. Pommaret CERMICS Ecole des Ponts ParisTech Paris France

A few physicists have recently constructed the generating compatibility conditions (CC) of the Killing operator for the Minkowski (M), Schwarzschild (S) and Kerr (K) metrics. They discovered second order CC, well known for M, but also third order CC for S and K. In a recent paper (DOI:10.4236/jmp.2018.910125) we have studied the cases of M and S, without using specific technical tools such as Teukolski scalars or Killing-Yano tensors. However, even if S(m) and K(m, a) are depending on constant parameters in such a way that S → M when m → 0 and K→ S when a → 0, the CC of S do not provide the CC of M when m → 0 while the CC of K do not provide the CC of S when a → 0. In this paper, using tricky motivating examples of operators with constant or variable parameters, we explain why the CC are depending on the choice of the parameters. In particular, the only purely intrinsic objects that can be defined, namely the extension modules, may change drastically. As the algebroid bracket is compatible with the

关键词： Formal Integrability Involutivity compatibility condition Janet Sequence Spencer Sequence Minkowski Metric Schwarzschild Metric Kerr Metric

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Solution of the Matrix Second Semi-Tensor Product Equation A ∘ l X ∘ l B=C

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Journal of Applied Mathematics and Physics 2024年第10期12卷 3261-3280页

作者： Hao Zhang School of Mathematics and Statistics Shandong Normal University Jinan China

In this paper, the solution of the matrix second semi-tensor product equation A∘lX∘lB=Cis studied. Firstly, the solvability of the matrix-vector second semi-tensor product equation is investigated. At the same time, the compatibility conditions, the sufficient and necessary conditions and the specific solution methods for the matrix solution are given. Secondly, we further consider the solvability of the second semi-tensor product equation of the matrix. For each part, several examples are given to illustrate the validity of the results.

关键词： Matrix Equation The Second Semi-Tensor Product compatibility condition Sufficient and Necessary conditions Vectorization

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Event-triggered distributed model predictive control for constrained linear system with random packet loss

Event-triggered distributed model predictive control for con...

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第37届中国控制会议

作者： Xiaoming Tang Shuang Yang Li Deng College of Automation Chongqing University of Posts and Telecommunications Key Laboratory of Industrial Internet of Things and Networked Control Ministry of EducationChongqing University of Posts and Telecommunications

This paper investigates the problem of event-triggered distributed model predictive control（MPC） for a set of linear local systems subject to random packet loss. We present a new model for each local system associated with a cost involving neighboring system states, in which a stochastic variable is adopted to describe successive packet loss in a random way. To achieve the stability of the global closed-loop system, a compatibility condition is imposed in the distributed MPC optimization problem. Moreover, an event-triggered communication scheme is constructed to determine whether the optimization problem is solved or not, which reduces the communication energy consumption and lowers the computational burden. Based on the linear matrix inequalities（LMIs） technique, we propose an algorithm by which the closed-loop stability of the whole system is guaranteed. In final, numerical example is given to illustrate the effectiveness of the proposed approach.

关键词： Event-triggered Distributed MPC Random packet loss compatibility condition

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