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An Overview and Recent Advances in Vector and Scalar Formalisms: Space/Time Discretizations in Computational Dynamics-A Unified Approach

机译:向量和标量形式主义的概述和最新进展:计算动力学中的时空离散化-统一方法

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An overview of recent advances in computational dynamics for modeling and simulation is described. The targeted objectives are towards a wide variety of science and engineering applications in particle and continuum dynamics of structures and materials which fall in this class. Starting with the supposition that in the beginning, the well known Newton's law of motion for N-body systems is given, and is a statement of the principle of balance of linear momentum, subsequently, using this as a landmark, firstly, the principal relations to various other distinctly different fundamental principles which are of primary interest here are established. Likewise, for continuum dynamics of structures, via the principle of balance of linear momentum, analogous developments are also established. Consequently, these distinctly different fundamental principles are shown to serve as the starting point for the various developments. Stemming from three distinctly different fundamental principles, we present recent advances in N-body dynamical systems, and also continuous-body dynamical systems with focus on numerical aspects in space/time discretization. The fundamental principles are the following:rnthe Principle of Virtual Work in Dynamics, Hamilton's Principle and as an alternate (due to inconsistencies associated with Hamilton's principle), Hamilton's Law of Varying Action, and the Principle of Balance of Mechanical Energy. Both vector and scalar formalisms are described in detail with particular focus towards general numerical discretizations in space and/or time for N-body and continuum-elastodynamics applications which are encountered in a wide class of holonomic-scleronomic problems. The formulations include the classical Newtonian mechanics framework with vector formalism, and new scalar formalisms with descriptive functions such as the Lagrangian, the Hamilto-nian, and the Total Mechanical Energy to readily enable numerical discretizations. The concepts emanating from the present developments and distinctly different fundamental principles inherently: (1) can independently be shown to yield the strong form of the governing mathematical model equations of motion that are continuous in space and/or time together with the natural boundary conditions; the various frameworks for the case of holonomic-scleronomic systems with the mentioned limitations are indeed all equivalent, (2) can explain naturally how the weak statement of the Bubnov-Galerkin weighted-residual form that is customarily employed for discretization with vector formalism arises for both space and time, and (3) can circumvent relying upon traditional practices of conducting numerical discretizations starting either from balance of linear momentum (Newton's second law) involving Cauchy's equations of motion (governing equations) arising from continuum mechanics or via (1) and (2) above if one chooses this option. The concepts instead provide new avenues for numerical space/time discretization for continuum-dynamical systems or time discretization for N-body systems, and consequently, they enable equivalences to be drawn from amongst the various frameworks under certain restrictions to provide a unifiedrnapproach and viewpoint. In addition, for the time discretization, focusing attention upon the class of Linear Multi Step (LMS) methods which are the most popular in research and commercial software, we particularly describe from a unified viewpoint, new avenues of discretization of the equations of motion via a new Total Energy framework. The resulting developments lead to improved physical insight, inherit computationally attractive features for developing algorithms by design, and new and optimal algorithm designs, while recovering most of the developments that currently exist from traditional and/or classical practices.
机译:描述了用于建模和仿真的计算动力学的最新进展的概述。目标是针对此类中结构和材料的粒子和连续体动力学的各种科学和工程应用。首先假设,给出了N体系统的牛顿运动定律,它是线性动量平衡原理的陈述,随后,以其为界,首先是主关系在这里建立了与各种其他明显不同的基本原理。同样,对于结构的连续动力学,通过线性动量平衡原理,也建立了类似的发展。因此,这些截然不同的基本原理被证明是各种发展的起点。从三个截然不同的基本原理出发,我们介绍了N体动力学系统以及连续体动力学系统的最新进展,重点关注时空离散化中的数值方面。基本原理如下:动力学中的虚拟功原理,汉密尔顿原理以及作为替代(由于与汉密尔顿原理相关的不一致),汉密尔顿的变化作用定律和机械能平衡原理。矢量和标量形式主义都进行了详细描述,尤其着重于N体和连续弹性动力学应用在空间和/或时间上的一般数字离散化,这在各种各样的完整-硬化问题中都遇到过。这些公式包括具有矢量形式主义的经典牛顿力学框架,以及具有描述功能的新标量形式主义,例如Lagrangian,Hamilto-nian和Total Mechanical Energy,可以轻松实现数值离散化。从当前的发展和固有的明显不同的基本原理中产生的概念:(1)可以独立地显示出支配的数学模型运动方程的强形式,这些方程在空间和/或时间上是连续的,并且具有自然边界条件;具有上述局限性的完整-硬化系统的各种框架确实都是等效的,(2)可以自然地解释通常用于矢量形式主义离散化的Bubnov-Galerkin加权残差形式的弱声明是如何产生的。 (3)可以依靠传统的进行数值离散的方法来规避,该传统方法是从线性动量的平衡(牛顿第二定律)开始的,该线性动量涉及由连续力学引起的柯西运动方程(控制方程),或者通过(1)和(2)如果选择此选项。相反,这些概念为连续体动力学系统的数值空间/时间离散化或N体系统的时间离散化提供了新途径,因此,它们使在一定限制下可以从各种框架中得出等价关系,从而提供了统一的方法和观点。此外,对于时间离散化,着眼于研究和商业软件中最流行的线性多步(LMS)方法类别,我们从统一的角度特别描述了通过运动离散化运动方程的新途径。新的全面能源框架。所产生的进展导致改善的物理洞察力,继承了设计上开发算法的算法吸引人的特征,以及新的和最佳的算法设计,同时恢复了传统和/或经典实践中当前存在的大多数发展。

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    《Archives of Computational Methods in Engineering》 |2011年第2期|p.119-283|共165页
  • 作者单位

    Department of Mechanical Engineering, University of Minnesota at Twin Cities, 111 Church St. SE, Minneapolis, MN, 55455, USA Department of Mechanical Engineering, University of Minnesota at Twin Cities, 111 Church St. SE, Minneapolis, MN, 55455, USA Department of Mechanical Engineering, University of Minnesota at Twin Cities, 111 Church St. SE, Minneapolis, MN, 55455, USA Department of Mechanical Engineering, University of Minnesota at Twin Cities, 111 Church St. SE, Minneapolis, MN, 55455, USA Department of Mechanical Engineering, University of Minnesota at Twin Cities, 111 Church St. SE, Minneapolis, MN, 55455, USA;

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