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力学与实践 ›› 2013, Vol. 35 ›› Issue (1): 1-7.doi: 10.6052/1000-0879-12-218

• 专题综述 • 上一篇    下一篇

曲面物理和力学:最佳基本微分算子对

殷雅俊   

  1. 清华大学航天航空学院工程力学系,北京 100084
  • 收稿日期:2012-05-29 出版日期:2013-02-08 发布日期:2013-02-07
  • 作者简介:殷雅俊,清华大学航天航空学院工程力学系教授,博士生导师. 1985 年在清华大学水电系水力机械专业本科毕业. 1987 年在清华大学力学系固体力学专业获硕士学位,同年留校任教至今. 先后获得宝钢教育基金会优秀教师奖、北京市和国家级教学优秀成果奖和北京市名师奖. 1993~1994 年,任荷兰Delft 大学客籍研究员. 1998 年获得日本广岛大学机械系工学博士学位. 2000~2001 年,任日本IHI 基础技术研究所海外研究员. 2003 年至今,主攻研究方向为微纳米生物力学、生物膜力学、生物膜纳米管力学、生物分形力学和昆虫仿生力学.
  • 基金资助:

    国家自然科学基金资助项目(11072125, 11272175).

PHYSICS AND MECHANICS ON CURVED SURFACES: THE MOST OPTIMAL PAIRS OF FUNDAMENTAL DIFFERENTIAL OPERATORS

YIN Yajun   

  1. Department of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing 100084, China
  • Received:2012-05-29 Online:2013-02-08 Published:2013-02-07

摘要:

曲面物理和力学中有两个独立的基本微分算子(即"基本微分算子对"). 本文综述如下主题:在所有的基本微分算子对中,经典梯度▽(···) 和形状梯度▽ (···) 的配对[[▽,▽]] 是最佳的. 具体内容包括:(1)基本微分算子对的形式并不唯一;(2) 内积的可交换性确立了[[▽,▽]] 优于其他基本微分算子对的"最佳" 地位;(3) 基于[[▽,▽]] 可以最佳地构造曲面物理和力学的高阶标量微分算子,因而[[▽,▽]] 是构造曲面物理和力学微分方程的最佳"基本砖块";(4) [[▽,▽]] 在软物质曲面物理和力学中普遍存在.

关键词:

曲面物理和力学|经典梯度|形状梯度|最佳基本微分算子对

Abstract:

There are two independent fundamental differential operators (called the "fundamental differential operator pair") on curved surfaces. This paper focuses on the topic: Among all fundamental differential operator pairs, [[▽,]], formed by the classical gradient ▽(···) and the shape gradient (···), is the optimal one. The following conclusions are included: (1) The paths for constructing the fundamental differential operator pairs are not unique. (2) The commutative nature of the inner-product of [[▽,]] is the basis of its optimality and advantage over all other fundamental differential operator pairs. (3) Based on the inner-product of [[▽,]], all higher order scalar differential operators for physics and mechanics on curved surfaces can be constructed optimally. In other words, [[▽,]]is the optimal "fundamental brick" for establishing the differential equations of physics and mechanics on curved surfaces. (4) [[▽,]] exists universally in physics and mechanics on soft matter curved surfaces.

Key words:

physics and mechanics on curved surfaces|classical gradient|shape gradient|optimal pair of fundamental differential operators

中图分类号: 

  • O302