Algorithms by design: A new normalized time-weighted residual methodology and design of a family of energy-momentum conserving algorithms for non-linear structural dynamics

A new approach termed as the hybrid displacement-strain-based normalized time-weighted residual approach is presented to accurately model non-linear structural dynamic problems. The focus here is restricted to all possible algorithmic designs within the class of linear multi-step methods that fall under the umbrella of the generalized single solve single step linear dynamic framework originally developed via the classical time-weighted residual approach involving a single solve within each time step as such designs are the most predominant in research and commercial software (Int. J. Numer. Meth. Engng 2004; 59:597-668; Int. J. Numer. Meth. Engng 2006; 66:1738-1790). However, traditional practices via classical time-weighted residual approaches fail to preserve the underlying physics and stability and do not serve the purposes of extensions to non-linear dynamic situations. In contrast, a new normalized time-weighted residual approach is proposed that naturally enables such extensions leading to the design of a family of conserving time operators for unconstrained conservative dynamic systems without resorting to enforcing energy constraints as in past practices. Consequently, the algorithmic stability (in the sense of energy stability) via this approach for non-linear dynamic problems is also preserved. For linear dynamic situations, it reverts to the classical paradigm. Only simple numerical illustrations are purposely presented to demonstrate the basic concepts.