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Algorithms by Design: Part III—A Novel Normalized Time Weighted Residual Methodology and Design of Optimal Symplectic-Momentum Based Controllable Numerical Dissipative Algorithms for Nonlinear Structural Dynamics

While in Part I (see [1]) and Part II (see [2]) of this three-part exposition we focused attention on non-dissipative symplectic-momentum conserving designs of time operators for applicability to nonlinear dynamics, here in Part III of this exposition we demonstrate how to further advance the theoretical developments and introduce controllable numerical dissipation via a novel time weighted residual approach we have previously described. The unique aspects of the algorithmic designs are such that when the numerical dissipative features are turned off, the resulting time operators readily recover the original designs of algorithms that are inherently symplectic-momentum conserving (we defer to those that are energy-momentum conserving elsewhere [3, 4]). The focus of the current work is on the theoretical developments of the new formulation termed the displacement based normalized time weighted residual approach for nonlinear dynamics applications. In particular, we provide extensions of the well—known Generalized Single Solve Single Step (GSSSS) framework, which comprises two distinct classifications, namely constrained U and V algorithmic architectures, and was originally developed for solving linear dynamic problems. Starting with the generalization of the classical time weighted residual approach, the GSSSS framework that was previously developed encompasses the general class of LMS methods represented by the single field form of the second order ordinary differential equations in time involving a single solve, and also covers most of the developments to date in the literature, including providing new avenues towards optimal designs of algorithms. However, the classical time weighted residual approach fails to adequately provide proper extensions to nonlinear dynamics applications. Consequently, the basic premise and argument that is herein advanced is that controllable numerical dissipative time operators designed for linear dynamic problems are very valuable, and are indeed the basis and can be readily employed as the basic parent algorithms, such that when implemented appropriately via the present new time weighted residual representation, they are now readily suitable for extensions to nonlinear dynamics applications. To demonstrate the basic concepts, we consider applications to the Saint Venant Kirchhoff material model simply for illustration, although the method can be readily extended to general material models. The numerical examples presented show that in contrast to the classical time weighted residual approach, which fails to recover the original designs of symplectic-momentum conservation when numerical dissipation is turned off, this new approach readily accomplishes this feature naturally and without enforcing any added constraints, and is the more appropriate way to design a particular class of controllable numerical dissipative schemes. It also leads to algorithm designs that yield fewer numerical oscillations in the energy and angular momentum in contrast to the classical approach, thus additionally confirming the improved effectiveness of the proposed approach. Further, we also show that amongst all the controllable numerical dissipative schemes considered in the sense of and under the framework of LMS methods in the single field form and involving a single solve and second-order time accuracy, the U0− V0 optimal is the preferred choice of this particular class of symplectic-momentum conserving based controllable numerical dissipative schemes since: (i) it yields least amount of energy dissipation; and (ii) it is ideal for any given set of initial conditions in the sense that it possesses the highly desirable attributes involving zero order displacement and velocity overshooting behavior. Simple numerical examples are provided that illustrate the fundamental ideas for applications to nonlinear dynamics problems.