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

Kumar K. Tamma, Jason Har, Xiangmin Zhou, Masao Shimada, Andrew Hoitink

Research output: Contribution to journalArticlepeer-review

40 Scopus citations


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: the 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 Hamiltonian, 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 unified approach 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.

Original languageEnglish (US)
Pages (from-to)119-283
Number of pages165
JournalArchives of Computational Methods in Engineering
Issue number2
StatePublished - Jun 2011

Bibliographical note

Funding Information:
Acknowledgements The authors are very pleased to acknowledge support in form of computer grants from the Minnesota Supercomputer Institute (MSI), Minneapolis, Minnesota. Special thanks are due to part of the contributions from Ms. S.U. Masuri as related to the symplectic-momentum and energy-momentum single-field form time integration developments.


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