Advances in Computational Dynamics: a Unified Approach

Abstract eng:
In this invited presentation, we highlight recent advances in computational dynamics and a unified approach for modeling and simulation of particles, materials, and structures. In contrast to the classical mechanics setting stemming from the Newtonian mechanics framework and vector formalism, or employing descriptive scalar functions via scalar formalisms such as the Lagrangian and the Hamiltonian, we additionally provide new and different perspectives with improved physical insight for enabling algorithms by design for modeling and simulation of applications related to computational dynamics. This is via a new scalar formalism employing the so-called Total Energy framework which involves a measurable descriptive scalar function, namely, the total energy. Firstly, to enable the modeling and simulation, the associated equations of motion for dynamic systems that are continuous in space/time are presented. The space discrete form of the equations of motion via the total energy representation is equivalent to and naturally represents the so-called semidiscretized equations of motion. Secondly, via the Total Energy framework which provides improved physical insight, and in conjunction with a generalized time weighted residual approach, we subsequently describe the theoretical basis and the notion of algorithms by design for developing a wide variety of algorithm designs for discretizing the time domain. The focus is upon the parent linear dynamic algorithms and designs which is a necessary first step in the design of algorithms for integrating the equations of motion. Most of the past developments over the years in linear multistep methods (LMS) following tradiitonal practices that are numerically non-dissipative and with controllable numerical dissipation features including new and optimal algorithm designs are an integral part of the developments. Thirdly, we describe a new normalized time weighted residual approach via the Total Energy framework which provides the necessary theoretical basis for properly extending the parent linear dynamic algorithm designs to nonlinear dynamics applications. It provides the theoretical basis that is inherently based upon and yields symplectic-momentum conservation or energy-momentum conservation for a wide variety of algorithms and designs, including covering those that have been traditionally derived from various different viewpoints.

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