On the Numerical Discretization in Space and Time: Part 2 - Total Energy Framework and Satisfaction of Conservation Properties of Discretized Equations of Motion for Linear Dynamics Problems

Abstract

For the modeling and simulation of computational dynamics applications with particular reference to holonomic-sceleronomic systems towards which this paper is restricted, it is important to ensure that the space and time numerical discretizations enable proper designs such that the fully discretized system also inherits the same physical attributes (such as energy, linear, and angular momentum conservation properties) as in the continuous case for conservative dynamic systems. Of the various computational frameworks (Newtonian, Lagrangian, Hamiltonian, and Total Energy) that we presented in Part 1, here in Part 2 of the exposition we particularly describe and advocate the so-called Total Energy computational framework for elastodynamics applications. This is because it is a self-contained framework in a natural setting which is computationally attractive. Consequently, it naturally enables the design of structure preserving time integration algorithms, and also inherits the advantages of readily ensuring proof of satisfaction of the conservation laws for the space discretized system. In contrast to Part 1, here in Part 2 of the paper we altogether discard the developments starting from the continuous representations to derive the semi-discretized equations. Instead, we describe yet another route directly starting from the supposition of energy conservation for the space discretized system. In particular, via the Total Energy framework, simply for illustration we describe the design of computational algorithms that are energy-momentum conserving within the scope of the particular class of LMS methods involving a single system of equations with a single solve within a single step representation, since these are the most popular in commercial codes. We demonstrate conserving algorithms by design for linear dynamic systems, which is a necessary first step and basis that is required for consequent extensions of the parent linear dynamics algorithms to general nonlinear dynamic applications, which we hope to disseminate in the near future.