Load aliasing—A new additional test concept for effective control of nonhomogeneous high‐frequency behavior in linear multistep methods

Abstract

This article introduces a new additional test concept and numerical property of implicit time integration methods termed as “aliasing” of the external load which must be considered during the algorithm design process. Where most numerical properties of such algorithms discussed in the literature pertain to the homogeneous components, aliasing reflects the tendency of the nonhomogeneous components to introduce errors into the solution in the asymptotic high‐frequency limit. An analytical technique is proposed to detect and quantify aliasing, defined in terms of the resulting representative solution. A theorem establishes whether a representative solution with aliasing can be bounded up to a constant by the external force values and thus treated as acceptable if the bound is sufficiently small. For the U0 family of algorithms within the GSSSS‐II framework containing several traditional and closely related methods, this is not an issue. However, the numerically dissipative GSSSS‐II V0 family of methods is shown to exhibit bounded aliasing with a constant roughly equal to 2, while the “structure‐dependent” Chang Family Method is shown to exhibit unbounded aliasing, which is not acceptable for practical dynamic computations. Numerical verifications are performed with single‐degree and multi‐degree of freedom problems with a series of selected algorithms which exhibit either no aliasing, tightly bounded aliasing, unbounded aliasing, or large bounded aliasing, and practical recommendations are given for acceptable aliasing of the load with special emphasis on MDOF problems such as those arising from finite element discretizations.