A variationally consistent framework for the design of integrator and updates of generalized single step representations for structural dynamics

Abstract

SUMMARYA variationally consistent framework leading to the concise design of both the 'integrator' and the associated 'updates' as related to the single step representations encompassing the so-called LMS methods for structural dynamics is described. The present paper shows for the ÿrst time, a consistent treatment involving both the 'integrator' and 'updates' that are inherent in the general context of designing the time integration process. Furthermore, the framework encompasses not only all the existing time integration algorithms that are dissipative and non-dissipative within the scope of LMS methods but also contains new optimal algorithms useful for practical applications-in the sense of accuracy, stability, numerical dissipation and dispersion, and overshoot characteristics of computational algorithms for time dependent problems encountered in structural dynamics.

Extension of LMS formulations for L‐stable optimal integration methods with U0–V0 overshoot properties in structural dynamics: the level‐symmetric (LS) integration methods

“…Present days again show the tendency to obtain, for stiff oscillating problems, the methods in single-field form with the use of all derivatives of solution in separate integration levels, because of more versatile set of parameters of such methods, see [15][16][17][19][20][21][22][23].…”

“…In this field, such implicit LMS methods for ODE sets in double-field form, as constant back difference formula (BDF) methods (see [18]), Gear integration algorithm [5], changing order and step size of BDF methods, had often been used. Present days again show the tendency to obtain, for stiff oscillating problems, the methods in single-field form with the use of all derivatives of solution in separate integration levels, because of more versatile set of parameters of such methods, see [15][16][17][19][20][21][22][23].…”

“…Most of these methods were recently encompassed within the unique one-step framework in a generalized single step single solve (GSSSS) algorithm created by Tamma and Zhou, see [16,17,[19][20][21][22][23].…”

Design spaces, measures and metrics for evaluating quality of time operators and consequences leading to improved algorithms by design—illustration to structural dynamics

“…However, techniques that can particularly identify these non-physical high frequency modes have not yet been developed to-date and such techniques have not yet been demonstrated for structural dynamic problems. After all, the Newmark average acceleration method is the most widely used implicit algorithm for structural dynamic problems and hence is used here simply for illustration (although controllable dissipative algorithms could have been selected and are expected to yield similar conclusions; see Reference [30]). …”

“…However, the CPU per time step of these two methods will be more than that of the central difference method because the solution of a system of equations is involved. Nonetheless, due to the large permissible time step (in contrast to the critical time step of the central difference method), although they are expected to be computationally more efficient for the overall time duration of the simulation, due to the non-linearly explicit nature of the FDEL algorithm, for a given problem with the same unconditionally stable time step size, the computational effort of the FDEL algorithm is expected to be significantly less than the Newmark average acceleration method for general non-linear dynamic situations (and from our previous experience and efforts [30] of comparisons with other dissipative and non-dissipative methods pertaining to LMS methods, this expectation also holds true). It is to be noted that in certain situations such as for highly non-linear dynamic situations, the LMS based implicit methods may have stringent requirements of time steps for accuracy considerations and to meet satisfactory convergence tolerances, and may be computationally more expensive than the explicit counterpart; however this is problem dependent.…”